Necessary but insufficient
It would help. It would more than help; I have a hard time imagining them being functional without it. The necklace of natural attacks or scorpion kama are generally necessary for Monks. That said, the necklace of natural attacks does exist (as do similar items in Pathfinder), and it’s not nearly enough to make the Monks good.
The Monk’s problems are problems of design: the people who wrote it evidently had no clear idea of what a Monk was or should do. Thus it receives a mish-mash of random abilities that do not synergize (and frequently contradict one another, see Flurry of Blows and Fast Movement).
To “fix” the Monk, one must first come up with a clear vision of what the Monk is supposed to be and do, and then most likely rebuild it from the ground up focused on that vision.
Or simply use one of the many classes that can effectively model one or more possible visions of what a “monk” should do, without any levels in the “Monk” class. The Cleric and Psychic Warrior are both Open Game Content and quite capable of fulfilling most roles you could imagine for the Monk, for instance. I’d argue that it’s entirely reasonable to treat a Barbarian’s Rage as “Zen Focus” and waive the non-lawful requirement (which I’d further argue is dumb to begin with). Such a “Barbarian” multiclassed with Fighter for combat maneuver mastery and perhaps taking Improved Unarmed Strike would make a decent monk. And if you have Tome of Battle, the Swordsage also does an excellent job.
Let's assume that the combat system is balanced as-is: that is, that the existing tools the system provides for defensive options in combat have the appropriate trade-offs between cost and benefit.
The basic tool to give an attacker disadvantage is the Dodge action. In addition to the obvious impact of having to spend an action taking Dodge (mostly: not being able to attack or cast a spell), there's a slightly subtler consequence. A creature has to choose to Dodge before the attacks start. This means the Dodging creature is gambling that the cost of forgoing some other action now will be worth the overall defensive benefit in the future.
Drop Prone and Give Ground instead give a defensive benefit now in exchange for a reduction in movement in the future. So the cost is paid after the benefit is received. Even setting aside the actual costs, this is a much better deal for the character! Suppose the house rule was simplified to just:
Desperate Dodge: Any character may spend their reaction to take the Dodge action until the start of their next turn. If they do, they cannot take an action on that turn.
This is much better than the regular Dodge, because a character is guaranteed never to use it unless they really need it. And they can plan around not being able to act on their turn. If this option existed, nobody would use the regular Dodge action.
Now consider:
Cautious Dodge: Any character may spend their reaction to take the Dodge action until the start of their next turn. If they do, they forego their move on that turn.
Cautious Dodge is much better than Desperate Dodge, which was already better than Dodge: it trades the major disadvantage of losing an action for the much more manageable disadvantage of losing movement.
Drop Prone and Give Ground are both generally significantly better than Cautious Dodge. They are slightly more limited (not affecting all attacks, having other minor drawbacks, etc), but their costs are generally lower as well. And much of the time characters only need to protect themselves against a single attack, and can mitigate or ignore the other downsides.
Given that these options are good enough to make one of the basic combat actions much worse by comparison, I would say they are unbalanced relative to the other actions characters have in combat. They remove one of the basic balancing mechanisms by allowing characters to trade immediate benefits for relatively small future costs.
Best Answer
Starfinder preserves the focus on melee-range combat.
There are a wide variety of elements that affects this, but ultimately there are 2 that dominate the range/melee issue.
First, Starfinder does not provide an easy way to "cover" an area with ranged weapons (i.e. to punish foes for moving through that area if they attempt to close to melee range). This makes closing to melee range a relatively trivial matter, unlike what would be the case in real life (where anyone attempting to run forward without cover, or at least covering fire, stands a good chance of being shot rather than reaching their opponent).
Second, melee weapons simply do more damage than ranged weapons, so there is a considerable incentive to close to melee range (especially against foes that would prefer to use ranged weapons). Here's a chart indicating the best average damage per attack (disregarding ammo usage, special abilities, ranges, etc) available at each level for each of the general categories of weapons:
Note that heavy ranged weapons just barely beat out one-handed (non-basic) melee weapons, and two-handed (non-basic) melee weapons are always the best choice in terms of damage per hit.