Right, it seems Porvoo ain't much of a beach town (unlike, say, Hanko, another Helsinki day trip destination). Here's what I found out.
Immediately north of the main tourist area, there's Maari (Maren) where you can "you can dip your toes in the water" according to tourist info lady. But this is river water, and she couldn't say if it's clean enough for swimming; at any rate the brownish water does not look inviting. (We didn't visit this place so I don't know what it's like exactly.)
Then there's lake Veckjärvi with a beach at Skaffas, some 5 km away from center (and another beach right across the water in Hasselholmen island which by road is 14+ km away). Another small lake with a beach in the same direction is Tervajärvi, some 6 km away. Local buses don't help much in getting to these places. (But e.g. with a rented bike these would be very accessible.)
The lady at the tourist info handed me this map of Porvoo beaches, listing only Hasselholmen (Skaffas actually), Tervajärvi, and Maauimala (Kokonniemi outdoor pool; see below):
As for sea beaches, there are many (as the link Mark posted shows) but apparently they are all far away (10+ km) and not really served by local public transport (which is very limited in such small town)—I'm not 100% sure, but this is the impression I got. Though there might be a beach in Hamari, close to where bus 2 runs... (Do post another answer if you know better!)
Anyway, for someone without car or bike, the best bet seems to be Kokonniemi outdoor swimming pool (Kokonniemen maauimala; address Jääkiekkotie 2), about 1.5 km from centre. You can easily walk here, albeit through far less charming areas than Old Porvoo. (Local bus 2 also stops right by the pool, but it runs infrequently (just once per hour on Saturdays) so walking may well be quicker. See the timetable at the stop next to the pool; note that the times mean departure from the terminus which is some 9 km away in Tolkkinen.)
This is where we ended up going on Saturday. So, not quite what I originally looked for (no sea or nice views!), but for swimming it's pretty good. Unlike what I first imagined, this pool is basically a small lake or pond (it's an old sand / gravel extraction pit), about 250 by 100 meters in size and 20+ m deep in the middle. It's surrounded by a sandy beach, plus some piers and jumping towers. It's free and open always (though the changing rooms, showers, toilets and kiosk are open 8-20 Mon-Fri & Sun and 8-18 Sat). That day wasn't quite as warm as forecasted, but the water at Kokonniemi was 21°C, probably much warmer than sea water.
Kokonniemi pool, photos by me
