It is odd for a major international border crossing to have a nice beach, but Sarpi's pebble beach and clear water are nice indeed. Despite the Turkish trucks lined up to cross over to Hopa, the setting is picturesque, with the beach and border crossing occupying a narrow strip of land before the topography of the village shoots directly upwards across jungle-covered green cliffs. Whether a swim here is your first or your last experience during your travels in Georgia, it is likely to be a memorable one. Swimming across the border would not certainly be appreciated by border guards, though.
The main eatery right by the border post and the beach is heralded by a lot of yellow umbrellas over plastic tables. It's a modest eatery and probably not the best introduction to Georgian cuisine, but some quick roadside food after a dip in the crystal waters of the Black Sea is likely to be enjoyable regardless of the quality.
Options are scarce if you are looking for accommodation within Sarpi itself, although if you make friends with a local you are likely to receive sincere offers of generous hospitality at their own house. There are good options just up the coast though—ask around after you've already left the customs office behind.
Quite naturally the next stop to the south is Hopa in Turkey. A ride on a dolmuş (shared taxi) should cost certainly no more than 5-10 Turkish lira. The Turkish taxi drivers won't speak Russian, if that's how you have been getting around, so brush up on either your Georgian or Turkish numbers and get ready for a lot of bargaining in Turkey!
If you are coming from Turkey, say your farewells to the dolmuş culture and say hello to the wonderful world of post-Soviet Lada taxis. A ride up to Batumi, where you can connect with the national transport system, should not cost any more than 15-20 Lari. If you do take a taxi into the city, your taxi driver may likely offer you a stop at a local cafe to try Ajaruli khachapuri—it's delicious and there is little reason to turn him down!
Better yet, though, be prepared to pay a little extra and ask your taxi driver to show you around some of Ajara's sites along the way. Or leave your travel plans for another day and take a taxi to one of the tiny resorts that dot the coast south of Batumi for more swimming and some good seafood.