I’m sorry to say that no, they do not. They will become a deeper shade of red, but the flavor won’t develop at all. As the berries get older, they will also get softer.
Because they don’t ripen, it is especially important to make an educated purchase – what you buy is what you get. You’ll find the best berries if you shop for them seasonally. In most places you can buy them any time of the year, but off-seasoned they may be shipped before they’re fully ripe, so for the best taste don’t buy them during the off-season. The berry should be firm, plump, and fully red with bright green stems fully attached. Green berries won’t have much flavor.
If you have the option, pick your own berries. This is the best way to know exactly what you’re getting and since picking them means they’re definitely in season, you won’t run into many of the problems you can from buying them in the store. Strawberries that are exposed to high temperatures and/or too much sunlight have a tendency to spoil, so the best way to help this already ripened fruit last longer is to store it in the fridge.