You can see spider mites. I have had somewhat extensive experience with them when I brought my dwarf meyer lemon indoors into a 'greenhouse' room I created in a storage room in my house. I kept humidity around 75% and heat around 80F. While I was able to rejuvenate my meyer lemon from sticks to a flourishing plant, I brought spider mites by the 1000's into the room.
There are two main types, and I won't get scientific about this to make it simple. There are reddish ones and there are semi opaque ones with a couple black spots on their back. A handheld 5x magnifying glass is enough to determine which.
You can also mist your plants with a sprayer and you'll see the webs at the interface of the leaves and branch as well as branch splits. You'll only see this if they have multiplied into somewhat significant numbers.
Spider mite eggs look like tiny pearls on the underside of the leaves. They don't lay them in any order like a ladybug might. The kinda look like the pearls that develop on young okra plants, but are significantly smaller.
The mites I had spread from the lemon tree to basil and quite a few other plants in my greenhouse. I ended up buying predatory mites that eat them. While the predatory mites will decimate a spider mite infestation, it takes too long for the predator mites to multiply into the numbers you need before a small amount of plants is overrun and destroyed by the pest mites.
The best way, I found, to get rid of them is to spray the leaves with a stiff mist and then just wipe them off. Or check each leaf and smash them. As Jal said, sprays risk the health of the plant. IF you get a decent compressed air water spray bottle, you can adjust the spray to simply spray them off. You'll have to pay attention to get them all off, though. But if you repeat this process over a couple days, hitting what you can day by day, you'll get the infestation under control.
The last problem you face is identifying how far they have spread. Spider mites, in my experience, don't really discriminate. They'll spread to anything they can reach - touching plants almost always get mite transfer. My infestation started on my meyer lemon. I paid most attention to it while not paying attention to the basil next to it. Once I had the meyer infestation under control, I noticed them on the basil...and then on the dill, and then the flowers we were growing...and they had even crossed the pot to the humidifier and were crawling all over it. They like the humidity. The more humid, the more they thrive when the temps are right.
As for predatory mites, you have to introduce them early if you want them to handle your problem or they will not replicate quick enough to get ahead of the pest mites - as I said. While they reproduce twice as fast as the pest mites, they can't catch up with a massive infestation. However, if you can get the predatory mites in early, they'll handle the problem in short order. Predatory mites don't eat plants, only mites. Once they kill all the available pest mites, they will start eating themselves until they are all gone/dead.