The Dead Sea is at the lowest point on earth and has such high salinity (due to evaporation) that you just bob about on the surface like a cork.
The most luxurious way to swim on the Jordanian side of the Dead Sea is at one of the upmarket resorts, where you'll pay from JD10 (Dead Sea Spa Hotel) to JD20 (Mövenpick Resort & Spa) for access to their private beaches and swimming pools.
Most budget visitors (foreigners and Jordanians) head for Amman Beach (3560800; foreign adult/child JD4/2, Jordanian JD1; 24hr), a clean public beach with good facilities, 2km south of the hotels. Locals generally swim fully clothed, though foreigners shouldn't feel uncomfortable here in a modest swimming costume.
A free alternative is the popular Herodus Spring, about 10km south of the hotel strip. Fresh (but undrinkable) water runs down its narrow canyon - ideal for washing afterwards. There's little privacy here so dress modestly.
Try to avoid Fridays and public holidays, when the hotels and public areas are very busy. Always take lots of water as the humidity and heat (over 40°C in summer) can be dehydrating and there's little shade. You need to shower after a dip in the Dead Sea to wash off the uncomfortable coating of encrusted salt. Don't shave before bathing!