Look. It depends on the beach. If you're in Sosua -- the main beach -- or Cabarete for example, you'll probably want a place to sit, lounge, park your bottom. Bringing a cooler just isn't the way to go there. You find a bar, restaurant (further down the beach at Sosua, West-ish, is cheaper) and or rent a whatnot, order from the nearest place and suck it up.
If you're out of town on a less touristy beach then by all means lug a cooler in your rented car. (Frankly the frigid beer that used to be the rule in the DR is no longer as universal as it was, so a cooler isn't a bad idea.) But at the same time on those beaches the beer won't be that expensive.