Hi, Jupiter. Sink strainers are both cleanable and replaceable. You might try washing with C-L-R first which will get rid of the build-up.
If that is unsuccessful, you can replace the fitting and tail piece. These are available at any hardware store or home center. You release the old fitting by unscrewing the nut under the sink. When you put the new fitting in, put a bead of plumbers putty around the bottom edge so it seals and doesn't leak. For most of the old ones that I have done, you usually end up replacing more of the old piping since it is usually in pretty bad shape also. When you are at the store, get everything down to and including the P-trap. This will probably save you a return trip.