This will unfortunately require a bit more work than you probably want to do when running the starter, but it's doable. You need a copy of this document. Take a gander at the second to last chapter and there you will find the encounter construction rules.
Basically what you need to do is find the encounter construction rules, reset the XP budgets to be for 6 players, and then rebuild each encounter with the appropriate number of monsters.
I would note that this is completely optional. You can run this straight up, let your party mow through it, and simply chalk it up as a learning experience.
One last suggestion if you don't want to change anything at all, is to adjust monster HP and damage. This is something I've experimented a bit with in my games with fewer people, but it's something that is definitely worth trying as monster HP and Damage are significant scale factors in 5e. Each monster has their average HP and damage listed. use a midpoint between their average and their high.
The Basic Rules are a place for you to start.
The Dungeon Master's D&D Basic Rules has a section on encounter building and modifying encounters (p.165-167) which will get you the same advice @Marius referenced in the DMG--at zero cost. Pay particular attention to "Party Size" on p. 166.
(The adventure contained in the Starter Set--The Lost Mine of Phandelver--does not contain notes on modifying it's encounters' levels.)
Note that the Basic Rules were revised in 2018. If you have the old Basic Rules, the referenced sections are on pages 56-58.
Take a look at Black Streams Solo Heroes, which has a lot of good advice for modifying adventures/campaigns/gm-style to suit the one-player experience.
Likewise, rpg.net has a good series of columns on Role-playing Duets that might give you some ideas.
But the best reference...
is to be up-front and honest with your players. Let them know you're not sure how things'll go and that you're very interested in their feedback. Some games are fun to play at the edge of survival, some players only want to play heroes who will embody the monomyth.
All of the references above are distilled wisdom of other GMs over the years, wisdom borne of experience. It's all good, but none of it's going to be better than the experience you all get playing together and discussing what you like and what you hate. Use these as references for ideas, but use your players and yourself as your guide.
There are a number of alternatives you have, let's talk about scaling down the encounters first:
Budgeted XP and encounters
A good way to adjust the difficulty of an encounter is to look at the budgeted XP for an encounter as intended by the published module and adjust it to your party-size. The guidelines for this is in page 82 of the DMG, or in page 165 of WoTC's DND Basic Rules. This involves calculating for each creature's XP multiplied by the Encounter Multiplier factor appropriate for the number of creatures that the party faces, this can be found in the same page of the DMG.
Page 83 of the DMG also takes into account the size of the adventuring party, saying to increase or decrease the multiplier by .5 according to the party's size. In your case, I’d increase the Encounter Multiplier by .5 because you only have 3 players and published adventures are assumed to be played with 4.
For example, if the published encounter comprises of 4 CR 1 creatures with a total of 800 XP value. Looking at the encounter multiplier table, 4 creatures yields a multiplier of 2x so you have an intended budgeted XP of 1600. Looking up the XP Thresholds by Character Level table, we see that the encounter is intended to be a Medium-difficulty encounter for 4 5th-level characters.
But since your party size is small, increase your encounter multiplier to 2.5x. So if you do not change the encounter, you'd have a budgeted XP of 2000, which is just over Medium-difficulty for a group of 3.
So, how do we tweak the encounter to be a Medium-difficulty for 3 parties? We know that for 3 5th-level characters, a medium difficulty encounter is budgeted at 1500 XP so we can simply take out 1 creature to reduce the budget to scale, since changing the number of creatures from 4 to 3 reduces the multiplier, we have a result of a budgeted XP for your small party of 1500 XP, perfectly adjusted.
That’s all well and good but what if the encounter is against only ONE really strong creature?
Modifying and Creating Monsters
Another way to balance the encounters is to create your own using the guidelines in page 273 of the DMG on Creating/Modifying a Monster (unfortunately, there doesn't seem to be a counterpart for these rules in the DM Basic Rules). This involves taking the essentials of a creature and whittling down its other stats so that it is adjusted to match the desired Challenge Rating. It’s all detailed in the DMG so I won’t put that in here.
Once you have your creature’s CR, look up its XP value and adjust it for the encounter’s difficulty, as mentioned above. This is your budgeted XP for the encounter.
Say, a CR 7 creature worth 2900XP (with a multiplier of 1x for 4 characters), would be close to a Hard encounter but way past Deadly for a group of 3. To make it only Hard, the budgeted XP should be 2250 XP which means the creature should be around CR 6 (5 if your player’s characters are low-tier). This is where the guidelines on Creating a Monster come in, you should modify the creature’s existing stats to knock it down a couple of CRs. This method is fine but tiresome unless you can find a resource tool somewhere on the net.
Alternatively, instead of scaling down the monsters, you might want to try scaling up the players by
1) Giving your players Hero Points
Hero Points are an optional rule found in Page 264 of the DMG, it works similarly to Bardic Inspiration and increases the chance of player success with a straight addition to any roll they make. This is a great way to boost your players up and making them feel strong and important. However, the players might feel like they’re being spoon-fed and might not like that so it would be best to talk to them about adding this option first, especially if there is a Bard in the party.
2) Give them Magic Items
Magic Items don't fit into encounter difficulty calculations unlike levels and Challenge Rating, but the more +1 weapons the small party has, the easier the encounters become. There are no guidelines on how much this affects difficulty, but personally, after running several high-magic campaigns for a normal-sized 6th-level party that has easily two combat magic items each, you can expect to increase their difficulty by a step (i.e. Medium encounters are Easy, Hard is Medium, Deadly is Hard).
Warning, though, you might want to avoid giving them +x Armor, as this can break the game with the PC's super high AC. Consider, instead, Adamantine Armor (or a similar magic item for light armor or unarmored PCs) which negates crits from hitting them, negating random deaths-by-chance. By the way, if you are designing a solo session, this is one of the best items to give them.
3) Add an NPC adventurer
You could simply add another adventurer of the same level of the party that you control. This is an added burden on the DM and it affects player agency a little but if you keep the NPC’s influence to a minimum, the players MAY allow this to happen. The extra muscle will certainly help but, like the Hero Points option, talk to your players about it first.
4) Allow the characters to take up apprentices/followers
Another way, related to the above method, is to allow your players to take up a 1st-level character to the party that they control as an apprentice or follower. This allows for a certain degree of help during combat and gives the players a fall-back character in case the higher-level one dies. Plus, seeing a player talk to (and even argue with) himself is golden (why should GMs be the only one to have to do this)!