Even if you're not staying in a hotel in Havana, it is still possible to find a place to sunbathe. Most of the hotels will let you go to their pool if you pay a cover charge.
I like sunbathing at the pools - there are sunbeds, no messy sand, and showers. When you get too hot you can go in the pool, and the waiters will serve you cocktails where you're sitting! These are the pools we would go to in Vedado...
Habana Libre, Calle M y 23
If you pay $10, you can use the pool. The $10 goes towards the bar. If you don't want to order anything from the bar, just pay $5. You can also borrow towels here with a deposit. They do a nice salad and decent cocktails. Also you can pop over the road and go to Waoo! which does amazing burgers!
The pool is nice, its not that secluded so you can still hear the traffic but its a great place to sunbathe. Sometimes there are creepy men there.
Hotel Nacional, Calle 21 y 0
This has one of the best pools. Its $15 cover. The pool is large and attractive.
If you're cheeky, you can sneak round the back of the Hotel in the gardens where there is another pool, where people go swimming. Often nobody checks here so you don't have to pay.
A bonus is you can sometimes get the Wifi at the pool here.
Hotel Meliá Cohiba, Calle 1ra
This hotel is more likely to have Cubans at it. Its a salt water pool! The cover is about $12, though you can try to sneak in, and they have a bar that sells things like fried chicken and pizzas.
Hotel Inglaterra, Paseo Prado
This hotel doesn't have a pool but it has a lovely roof terrace with live music in the evenings and perfect view of Old Havana. Its nice to go here for a drink.
There are other hotels in the Old Town, and if you go in and ask they'll let you to the pool. Just make sure they don't rip you off!