They are definitely spider mites! They usually go away with some humidity so maybe you can take it outside on warm days and spray it with some neem oil (Neem oil is not toxic to humans or pets) mixed with water and a tiny drop of soap. That and a rainy day might be enough to slow them down a little. Just make sure that when you spray the neem oil its not in the middle of the day because the oil might make the leaves burn with the strong sun. Also Neem oil smells bad so make sure you spray it outside and only when temperatures are above 50F (apparently it begins to harden at 50F).
Spider mites area a very common problem for palms indoors.
Good luck!
-Alex

Yuck I'm going to take mine out this week on our warm days and spray mine!! I'm not an expert, but I have quite a few palms indoors, my majesty palms do well with NO direct sunlight, i learned they like bright indirect light but not direct sun. Just FYI

I have a couple majesty palm's potted in doors as well. When they start to brown, I either cut the tips off of each falm leave, or cut the whole thing off when it all turns brown.

I noticed that these things need watered a lot for being in 75 degrees the majority of the time. I give it a big watering about once a week and let it dry out a little in between waterings.

Good advice to give it a spider mite treatment. Also, these palms like "some" sun, but not for > 4 hours. They like filtered sun-light and some direct sun. The plant looked a little dry - is it too close to a heat source or drying-out too much?

Thank you for the responses! I will see what I can do about those mites, and try to keep the palm happy.