There's no public pool (that I noticed) in Bovec (that I noticed anyway) and the Soča generally looks to be too fast flowing to practically and safely swim in. Where nearby Bovec (preferably) and beyond that in the broader Soča valley can I go for a swim?
Preference would be for outdoor places with at least some incident sunlight to warm up the cold water.
Soča is fast flowing only in it's upper part (from Bovec upwards) where people mostly indulge in whitewater canoeing, rafting and other sports. From Bovec downwards the river slows down and is great (and safe) to swim in.
Generally there are many places to swim - even upwards of Bovec where water tends to be colder and swimming sites can be less accessible - but the best places to relax, sunbathe and swim are between Kobarid and Tolmin on the numerous gravel banks. As you drive by the river you will see a lot of great spots - just pick one you like :)
Here are a few popular locations:
While the answer from Primož is good, I thought I'd add a couple of places I had found:
Photo by CMaster, released under CC BY-SA
A 5 minute drive or probably 20 minute walk or so outside of Bovec, and a popular sight-seeing spot. There are two pools under the fall, one of significant size and depth making for a good swimming location. The water is however, really quite cold, and the position of the waterfall in a hollow means that is receives relatively little sun.
Just upstream of where the Lepana flows in to the Soca, is Kamp Soca. On the opposite bank to the campsite is a large parking area, and a gentle beach that leads in to two areas suitable for swimming, one of them even allowing some jumps from height as it is at the very end of the Velika Korita ("Big troughs of the Soca"). This spot gets very busy in summer however, with both passing traffic and residents of the campsite.
