My handwriting got worse each time. So I don't recommend that.
I think you have to want it to improve, and to value it. And then you have to practice with that in mind. At that age, I was unable to see a reason to improve it, and it was a lot of work, so I didn't put any actual effort into it beyond that torture they were putting me through. (It hurts my hand to write.)
Lately, though, I've considered finally improving it. I think improving it will involve re-learning how to hold a pen and move my arm/hand while writing. They way I do it is basically guaranteed to produce bad results, and I should approach it more like art than utility.
That all said... I still haven't found the energy or resolve for it.
Good luck!
This is a decent tactic if, like you said, you want to improve your handwriting. You have to want it. You have to go slow and form the letters perfectly every time so you develop that muscle memory. I printed pages of a single letter in the font I wanted to imitate and traced the letters. It's a great activity to do while you're sitting in front of the TV.
edit to clarify: the decent tactic is to re-learn how to form your letters IF you want to improve your handwriting. DO NOT force your kids to do so. It will be counter-productive because they won't focus on forming the letters. They'll speed through it to get it done and wind up with worse handwriting than before. As wccrawford and I both said: You have to want it.
Mostly it's just slowing down. Write a little slower and pay more attention to how your words line up and spacing of letters. Read over what you have written and see which letters are hard to read. For letters and numbers that are ambiguous try adding something like a slash through zeroes and a dash through sevens.
If you write a lot this way people will start thinking that you had a computer fill out forms. It also helps a lot if you buy a good pen like a G-2. I find bigger nibs like a 1.6 keep my lines more even.
If he doesn't want to learn the Palmer method (what you probably learned if you grew up in the US), then perhaps Getty-Dubay?
There are a lot of penmanship videos on YouTube. I fell down that hole one day. They talked about picking a font a training to write in that font. But the kid would have to find the idea of picking the font they will write in compelling enough to train it.