The separation is not only near the faucet splash zone area but also further away near the wall.

Dig the old caulk out with something like an old credit card. Tough enough to remove it, but not scratch the surfaces. Clean it with acetone on a paper towel, then apply Axiom Translucent White silicone caulk. Spritz the bead with Windex, spit on your finger, and remove the excess. When you recaulk the backsplash/wall, use paintable caulk please.

Once done, make sure you wipe the counter and backsplash every time you use the sink.

If you are fussy consider this technique. Clean the old caulk. Then buy the 3M blue masking tape (which removes more easily than regular masking tape) and tape the countertop and splash leaving only a very small amount (1/16") of the counter and splash exposed. Then apply a thin bead of caulk to the joint. Now take anything with a 90 degree corner (I usually use a wood shim) and run the corner along the bead which pushes the caulk into the joint. Then carefully remove the tape which will have excess caulk on it. If you've never done it this way the tape will probably have a lot of caulk on it. After tape removal the caulked joint looks perfect. This approach is time consuming and you have to be meticulous about applying the tape but there is no better way to do it.

Obviously not the cause of your caulk issue, but a 35 year old condo can certainly settle ... sometimes with disastrous consequences.