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I'm kind of worried it would make some kind of 'bad' impression in my first day. Or am I overthinking?

Using crutches can happen to anyone, you should not feel bad about it.

If any, I suggest you write an email or similar to your new boss, explaining to him/her about the accident and the crutches, but that you will be there at work as agreed. This way your boss will be aware of the situation and able to act accordingly (that is, to help you with your needs given these crutches).

You are overthinking this. Accidents happen, and any employer worth working for will understand. If you're good-humored about it, you might even be able to endear yourself to them and leave a good impression on your team.

Yes you are overthinking. No one should think less of you, and unless your job is champion skier or Himalayan Sherpa or In a role that requires you to stand or walk most of the day (which may cause some issues for you doing your job), I don't see it being a problem.

I've gone to work as a software engineer on crutches for a few months due to a sports injury. You'll feel very self-conscious about it, and that's natural. I'm guessing that's probably a large part of where this question is coming from. There were some things I couldn't do for a while but if, like me, 90+% of your job is desk work, and this is temporary, I can't see any reason why it should be a problem.

Honestly, the biggest impairment for me (which I never thought about when able) was the inability to carry anything. Get a sealing water bottle, and that will solve a lot of problems.

I would highly suggest you get a written clearance from your doctor to return to work (for desk work of course). You didn't say where this job is, but I know here in the US my company's HR department wouldn't let me return without it. If you can reassure your new employer you have that and are willing to present it, there shouldn't be any problems.

If anything, this will work to your advantage.

You will have a guaranteed topic of conversation with your coworkers, since everybody will want to know how you did it and how you're feeling. Adopting an attitude of cheerful determination will also impress folks.

Plus, you will be the object of some sympathy, as long as your condition does not last for too long.

From another perspective -

You have a disability.

Albeit a short term one, but a disability nonetheless.

You didn't say where your employer is, but if it's in any of the English speaking countries, then it's very likely there is some kind kind of employment law in place to protect you from any kind of discrimination based on your disability. That means your employer cannot treat you any differently (in a negative way, at least) than any other employee. In fact, they may even be required to provide certain additional accommodations specifically for disabled employees. An injury requiring crutches probably won't limit your mobility all that much so it's unlikely they'd really need to change anything to accommodate this specific ailment, but still, they will be aware of this possibility and may have already added automatic doors, elevators, ramps, etc. for other disabled employees, and so you'd at least benefit from those improvements.

To be honest, when they see you come in on crutches, your employer is probably going to be more nervous about themselves making a good impression on you than the other way around. After all, they now have all the aforementioned additional responsibilities to live up to, at least until you get better.

All that said, I would not expect anyone to think you're making a bad impression anyway. As others have said - accidents happen. It may even be a nice little icebreaker, giving you something to start a conversation over with your new team mates, as long as you're comfortable chatting about whatever happened.

Been there, done that, didn't even get a t-shirt.

Sure, you'll get some strange looks and people will be curious, but it's nothing to worry about unless your job depends on you being to walk without crutches in which case you should contact them and suggest you start a month later after you no longer need the crutches.

Current job, I seriously bruised my ribs and had a bad wound on one foot that made it impossible to wear shoes for months while it healed just a few days before I started. People were surprised to see me like that, but after a few days everyone knew and that was it.

Biggest problem if you're on crutches will be getting to your work site, count on needing more time than normal as you'll be slower stumbling across train and bus platforms than you would normally (and if you'd normally take a car, that's even more relevant, as public transport tends to take more time than driving, and many office estates don't have that great a bus service running to them).

I do agree with the other answers on here but I just wanted to add that if there is a really lurid/unprofessional story behind how you hurt yourself. You might not want to share that with your boss on your first day.

You can't make a bad impression by having an injury that requires special accommodation. Even if you could, what are you going to do about it? It's not like you can just decide to to be able to walk safely unassisted.

You're thinking too much. Just go to work. It's a job, not a religious calling. Absolute worse case scenario is that your boss breaks the law and fires you. If that happens you can just go find another job, and you can probably sue your former employer.

I started writing a comment and accidentally wrote an answer.

There is absolutely no sane reason this can affect your career at all. Your will be seen as a worker that is willingful to sacrifice their own comfort in favor of their job done. If there are issues for you to get the contract it will give you one message: The employer is not worth your effort at all.

I had a serious accident right before I was about to start interviews. I had to postpone the first one for a week to be able to perform basic tasks like eating and such.

I got the job and the only odd looks were, and are, followed by a comment "He almost broke his neck and in two weeks he was succesfull at the interview."

Based on your question, I'm assuming you either had a phone interview, or you met them face to face without the crutches. I'm with Fattie and advise you to say nothing at all especially if you didn't have the crutches during interview. Simply make a joke on your first day like, "Yeah I had a accident and twisted my ankle. The doctor advise I use crutches for a month and I'm starting to feel better."