Left Crank came off
 

Last week I found a bike and decided to try to fix it up. It is a Bridgestone 200 (1985 ish). Today I did not first longer ride on it after overhauling the bearings (25 miles). Towards the end of my ride, my left crank came loose. I tried tightening it on with pliers but I wasn’t able to make it tight without my wrench. It came off 2 more times before I got a ride home. IT IS SQUARE TAPER. Should I replace the left crank or do I need to replace the the axel that goes through the bottom bracket? What are some budget options for just a left crank 170m? Is my crank worn or could it be the axel that goes through the bottom bracket or something else? Do i need any other measurements for a new crank besides length?

When a crank is ridden loose for an appreciable distance the taper in the crank is often distorted to the point where it will not stay tight; the spindle, being harder steel is usually OK.
Besides the length you will need to match the orientation of the taper, either "square" or "diamond". There are also JIS and ISO tapers, yours is parbably JIS: https://www.sheldonbrown.com/bbtaper.html

Try cranking it down good and tight and see if it holds. If not replace it. What do you have to lose?

Could this damage the spindle comes undone?

As disbrantjr said, the steel BB spindle is unlikely to be damaged by the softer aluminum crank arm. You really don't have much to lose but I'm betting the crank arm is ruined. This happens often enough that shops I've worked at used to keep generic left crank arms in stock. It's an economic repair if you aren't too picky about the crank arms matching side-to-side. FWIW, the torque spec for that crank arm bolt is usually around 30 ft/lbs. If you don't have a torque wrench imagine a 30 pound weight hanging off the end of a foot long wrench. It's more torque than I'd use if left to my uninformed judgement.

Take a close look at the crank around the square taper hole. Are there any cracks ? I had 1 come loose because it cracked.