Grandma's pearls, how to care for them?

Hey guys I have a strand of pearls here from my grandma and maybe even possible my great grandmother depending on what they date to exactly. 1930s, 40s, 50s? Anyhow they aren't in great condition as seen in the photos here. It looks like dirt and grime but may be damage... What is a safe way to clean pearls? Any help is greatly appreciated. Marking is as reads, "Sterling S"

 

That's damage, probably from hair lacquer. Get a new Sunshine cloth and give them a gentle wipe over. The best thing then is to wear them next to your skin.

 

I don't know if spartcom wants to wear them next to his skin, but keeping them in a pouch made of soft material, or a box lined with soft material, is fine too.
These are costume pearls, imitation. But the care is the same as for real pearls, always be gentle, use soft methods such as wiping, as Obb said, and don't soak in water. The damage won't go away, but this isn't the only vintage costume pearl necklace with damage.

 

If you want to test if a pearl is real rub it across your front tooth. If it feels gritty/scratchy it is real (or a cultured pearl) it feels horrible but works.

 

Real pearls and the more expensive costume pearls are often individually knotted, there are knots in the cord between each pearl.
The raised edges around the holes are an indication that these are not real pearls:

 

My take is that these are costume, real pearls have a slight variance in their surface iridescence. The tooth advice is a really good one and easy to do, I have used it myself when I was not sure. Sometimes you get a nice surprise, sometimes not. From the images here, you can see the wear and some losses of the surface coating near the bead hole, which is quite normal for it's age. The clasp looks like 1950's type of a good quality necklace.
I would gently use a dry soft brush, and buffing cloth, if you want to wear them. It matters to have something handed down from your family. This is an important keepsake, not matter what it is IMO.