I have taken my bike to many different locations and they all seem to give me different answers. My bike is an 87 Bridgestone 550. I really enjoy the bike because its a beautiful and rare freak of nature. I'm 6'4'' and super lanky making the 63 cm frame perfect. Long story short I bent the rear drive side dropout closed, and not thinking I grabbed the closest wrench next to me and I instantly pulled the piece back out cold. The piece now has a small crack next to the tension screw. My question is what would be the safest most cost efficient solution to my problem? The bike is my favorite commuter and I would like to not to have to scrap the old girl. Thanks for reading my post.

The only "safe" - as in entirely beyond criticism - way to fix that is to have the dropout replaced. As it seems to be a steel frame, its doable enough. A partial repaint and the bike is as good as new.
No drama at all to a frame builder, a modest challenge to a generic metal worker.
Someone not working bikes regularly would need to create some sort of jig to get the alignment right.
Apart from that metal is metal.
It'll cost some though, particularly if you want a blend-in paintjob.

Me, I'd remove the axle adjustment screw, grind a V-shaped groove along the crack on both sides. Stick an axle sized bolt at the open end of the dropout, use some washers and a nut to maintain alignment. Then use whatever arc welder that was handy to run a bead through the ground out groove to fill it up. A quick swipe with an angle grinder to get the dropout face flat and call it done. Maybe a dab of primer.
With luck, the adjustment screw would still fit. If it didn't, I'd try to retap it. If that didn't work I'd just consider it a "war wound" and do without it.

dabac's answer is pretty spot on. As the drop out's axle faces are pretty chewed up I'd vote for a replacement. Have a full assessment done on the rest of the frame's condition and consider how you'll want the paint to be dealt with before pulling the trigger for a repair. Andy.

This is a plausible repair option. Either welding as you suggest, or brazing ought to work. I'd get rid of the adjuster bolt completely, though, and fill the hole with brass so it won't act as a stress raiser again. You don't really need those adjusters, after all; they're just a convenience in getting the wheel centered and the holes they require in the dropouts are a common failure point as the OP has discovered.

One other thing, after you have the dropout replaced or repaired by welding or brazing, have the rear derailleur hanger check for, and if needed, realigned. It's not as critical for friction shifting but good alignment is required for proper indexing. I agree the wheel alignment bolt is a non-essential convenience and would make no effort to save it.

And be more careful next time. Looks like you opened the dropout further than you should have.

+1, have the dropout replaced

If you stick with index shifting you can have both replaced
and fit in a Vertical Dropout, pair.

Index shifting likes the wheel to always be in the same place... consistently.

Welding or replacing the dropout would certainly be a definitive repair, but you might consider just riding it while keeping a close eye on it. There is virtually no force on the cracked area and it does not support the primary load. Of course keep your axle nuts tight and monitor.

agree. ^^^

I should have said "There are very low forces on the cracked area . . . " mostly from the deraileur, and they get transferred to the axle almost directly. It's not in the structural part of the frame.

i STILL agree.

I guess I'll just have to accept it. (sigh)

I really appreciate all the input I received from everyone. Yall make it seem must simpler than I initially thought it would be. I took her for a spin today and I realized I need be more careful. Thanks for everything folks.

+1 lose the screws and fill it with weld or brass.

Oh, and align the hanger when you're preparing it to be welded or brazed, so it only needs minimal tweaking afterwards.