Should I give some money to a coworker?

There is a difference between "giving some money" and "loaning some money." If you want to "give some money" to your coworker, that's up to you. It's a nice thing to do if you can afford it. When my coworker didn't have enough money to put gas in her car and payday was still a couple of days away, I gave her a $20. I would have NEVER loaned her any money, though. Loaning money is a business transaction and she was simply not credit worthy. 

Well first question would be. How are you financially? If you yourself are not doing great you may consider alternatives or only give him maybe a smaller amount $100 instead of co-signing a loan etc. you don’t want to overextend yourself and get into the same position. You can also sit down with him and help him apply for unemployment again, he may have done something wrong that rejected him or he just may not know how to do it. A lot of states have been reporting a lag in responding time and their websites being down due to the large demand. A go fund me page is also an option, you or he can set it up and people can donate some money so you don’t have to do it all on your own, but have a community of people contributing $5, $10 or whatever they can spare. That way you’re helping but also doing it in a way you don’t have to bear the brunt of it or overextend yourself during this difficult time.

Then you should know that if someone is in dire straits (not straights) and they can't make it on what resources they have now, they are very unlikely to have them later--and very unlikely to pay back any loan. Even if you get him to sign a promissory note, you will be unlikely to get paid back. you've already gone through this--why do through it again?  Either make it a GIFT--without expectation of being paid back, or just simply don't do it. It's up to you. It's a kind gesture, but it can backfire on you pretty harshly if you don't make it absolutely CLEAR to the guy what you're doing--and that means either you simply GIVE him the money or don't. But don't expect to be paid back. 


And consider this: if a person has no resources when bad times hit, there is usually a good reason. Bad management of one's finances is the primary cause. There was nothing set aside for hard times or lean times. It's a pattern. And it's also a bad investment. 

Up to you, but don't count on his being able to pay you back.

Tell him you got burned in the past but maybe he can couch surf at your place for a bit.

First of all, there are people who I would definitely loan or give money to, and maybe there are those close to you that fall into that category.  But for the very reason that you're asking this question, I'd say don't loan money.  It will almost certainly break your friendship.  If you want to give him money, that's up to you, but it will almost certainly be only a temporary fix.

What you can do, if he is amenable, is help him in a way that doesn't impact your finances.  You could advocate for his unemployment, if the reason isn't sound.  You can refer him to job leads.  In my area, there are openings at grocery stores, for example.  If he loses his housing and has to live in his car, you can offer your shower and kitchen periodically.  You could refer him to a local food bank.  If you have skills, you can maybe teach him some things that will lead to a new job.

That's your decision. 
I myself struggle to pay bills, so recommend that somebody needing money try + sell things which they don't need or want anymore. Or as it turns out I need help cleaning my apartment .. If it's someone I trust in my home a loan can be done that way. Say like if I pay somebody 10 dollars an hour and they need 100, they have to work 10 hours to get that money from me.

people will always remember the sediment of a gift in hard times...if you dont want to give him money then partial/or whole pay a utility bill for him anonymously, gift him food....Sorry you got burned thats really hard to recover from. Maybe this is your time to heal from that..Be well my friend.  

You asked if you should "give" him some money.   Sure, if you want to, but if not, that's ok too.   Your money is yours to decide what to do with.

Note that "giving" someone money is not the same thing as "loaning" them money.  If you give him anything, it's a gift.  Do not expect to receive a single nickel of it back.  

What I've learned from life is to be compassionate with other people, so yes--offer him the loan.

What I've learned from watching TV reality court shows is that one should never give a loan without a written agreement signed by both parties.  So, unless you don't care about repayment, put it on paper.