Is it possible to get a bass boat trailer with brakes, like on a horse trailer or other type of trailer?  

Yep you can get surge brakes or elect. brakes. But when it comes to trailer brakes they are a pain in the butt. 8-)

I can imagine, but my little Hyundai Santa Fe can only pull 1500lbs...   :'(

Try to stay away from electric brakes. They are the cheapest way to add brakes to a trailer but a lot of them give problems when backed into the water. If you're going to add brakes, it cost more but the 10" disc brakes are the way to go.

I said 10" but that's going to depend on what size wheels you're running, if there's enough room for that much brake and  how much weight you're towing.  You might need to go down to a 7"

Electric brakes are cheap and easy it install but it might be a point that water in them might cause some problems. A dealer could tell you more. OR--you could do like Fred Flintstone and drag your foot. 8-)

I did some searching around, and it looks like a lot of manufacturers offer different breaking systems as options on their trailers... Thanks for the advice!