Boots of Striding and Springing add 10' to "base land speed". The swim skill provides you may swim at up to half your "speed" (as a full-round action) or at a quarter of your speed (as a move action).
Some things like Haste specifically say they do increase swim speed, but the Boots only talk about land speed and jumping.
Do the Boots increase the distance that a character can swim in a round?
The reference in Haste to Climb, Swim, etc. speeds is to actual speeds listed in a stat block (like a Spider's climb speed or a Leech's swim speed).
I don't think Haste has any direct impact on your rate of travel re: Swim or Climb checks via the skill. However, it does indirectly by enhancing your base land speed (which is what is used to calculate how far you can travel using those respective skills).
I can see the argument that the Boots are intended to increase your ground speed only. But, I'd allow it. The Boots explicitly increase your base land speed. Base land speed is what you use to calculate how far you can swim or climb using those respective skills. So I think it works.