Bear with me, this may seem like three questions, but they are so tightly related, I think that they need to be answered together as they are essentially all based on one ruling.
The PHB states:
Every character and monster has a speed, which is the distance in feet
that the character or monster can walk in 1 round.
Boots of Speed entry in the SRD states:
the boots double your walking speed
This is all well and good. My 6th level monk has a base speed of 45. When using the boots, his base speed is 90. Here's where it gets confusing. He wants to climb a wall or swim a river. The boots specify the modifier is only to "walking speed". However, the PHB says under "Climbing, Swimming and Crawling":
While climbing or swimming, each foot of movement costs 1 extra foot
(2 extra feet in difficult terrain), unless a creature has a climbing
or swimming speed
Without the boots, he should swim at a speed of 22" (45/2). Since the boots explicitly state "walking speed" not "speed", is my base movement while swimming 22" (my speed/2) or 45" (my walking speed/2)?
Furthermore, how does that figure into combined movement? If I walk for 30 feet, I should have 60 feet of walking left. Is that 7 feet of swimming (45 speed – 30 feet moved)/2, or 30 feet of swimming (90 walking speed – 30)/2?
And finally, since these are all very closely related: If I am playing a race or using an ability that grants me a swimming speed of 30, do the boots affect that? It seems nonsensical that the mere fact of having a swimming speed actually makes me slower than the more favorable ruling above (45 vs 30).
Best Answer
The Boots of Speed WILL Allow you to Swim a bit Farther
Official Ruling by Jeremy Crawford
Speed is defined as Walking Speed
When you see the speed on your character sheet or in a monster stat block that is walking speed
Also
So if your character has 30 ft speed, that is a 30 ft walking speed. If they have a 30 ft walking speed they have a 30 foot speed. They are defined in the game to be the same.
Your Walking Speed Doubles With Boots of Speed
When you put the Boots of Speed on, your walking speed doubles. Per the above, that just means that the stat called "speed" on your character sheet gets multiplied by two. In the example above the OP's speed/walking speed started at 45 then becomes 90. This is his new speed.
You can use ANY AND ALL Speeds you have to Swim
Jeremy Crawford says:
This means you can use your fly speed to swim. And your climb speed. And your burrow speed. As long as it is a speed you have, you can use it to swim. Of course, any of these that are not a swim speed will be used at a penalty (each foot of movement costs 1 extra foot), but you can still use them. Obviously this includes walking speed which is the default for most PCs when they go to swim.
Addressing the Specific Questions
Does OP's example character have >=1 speeds? Yes.
What type of speed(s)? A walking speed.
What is their current walking speed? 90 feet.
Can you use walking speed for swimming? Yes.
How far can they swim? 90/2=45 feet.
Simple as that.
In the example above, OP has a walking speed of 45. When he puts on the boots his walking speed becomes 90. THAT IS HIS SPEED. That is how far he can move in a turn. And according to the rules and JC's clarification above you can use that speed to swim.
To those that doubt this, what other number would the character use? His speed is 90 and he has no other types of speed. There is no other number he could use.
So, you would be able to swim 45' (90 feet / 2).
Combined movement
If you walk for 30 feet, you should have 60 feet of walking left as you said. You would then be able to swim for 30 feet (60/2).
Multiple Speeds
If I am playing a race or using an ability that grants me a swimming speed of 30, do the boots affect that?
Not at all.
Jeremy Crawford also said:
Since it specifies walking speed, it only increases your walking speed with no effect on any other speeds you might have.
Example
Just to clarify things let me make an example where your character has a 45ft speed (walking) and 15 ft swim speed.
Putting on boots of speed would mean that you now have a 90ft walking speed and a 15ft swim speed. You could then swim 15ft using just your swim speed. After that you will have 75 feet of movement left since:
So with the remaining 75 feet of movement, you would be able to go 75/2=37 feet further in the water. This means you would have moved a total of 15+37 = 52 feet in the water.
What is the Purpose of Specifying "Walking" Speed in the Item's Description?
If the item's description had been written "the boots double your speed", then all of a creature's speeds would be doubled (walking, flying, swimming, etc.). This was clearly not the intent.
So, by writing "double your walking speed" they are specifically indicating that those other types of speeds are not affected by this item. Which makes sense since they are boots and not flippers or jet engines. It does not restrict how and when that walking speed can be used though. If they had intended that, they would have added language to that effect.