I am renting an apartment and the circuit breakers are located in the downstairs apartment. Do I have a right to have access to them in case one of them trips? There is an exterior basement door, but the neighbors always have garbage stacked on top of it. The landlord has told me that I can go in to reset a breaker if I need to, but is it against the law?

In order to protect yourself, I would suggest having some sort of written agreement between the downstairs neighbor, yourself and the landlord concerning you having access to the circuit breaker cabinet. If I were you, and if I didn't know the downstairs neighbor all that well (don't know what your situation is concerning that), I would be very reluctant to have a key to the downstairs apartment. If I didn't know the other tenant all that well, it would be possible for him or her to accuse me of various things if I did have a key.

Around where I live, normally, a landlord can just go into an apartment if the renter is not there. Another tenant would not have the same privileges and there would need to be some sort of written agreement for the protection of all involved.

If there is garbage piled upon a basement door, then I think that would be an issue that city building inspectors would not be approving because doorways need to be clear of rubble and other items so that people can leave freely in an emergency.

I will agree, the trash is one issue that you need to address with the landlord and/or the city inspectors. This is a health issue and should not be allowed.

Unless you have it in writing, I would not enter the other apartment, but unless you have local codes that require the box in your apartment, this is not that uncommom in converted rentals.. Is this apartments legal ? In that do they have a permit for apartments

Also, it probably possible that the person in the other apartment could cause a breaker to trip, shutting off some of your power. This is probably more common in apartments then we realize. I don't know if electrical codes cover this but they probably do.
I wouldn't let myself into somebody else's apartment for any reason (unless it was a friend and they asked me to feed the cat while they were away). How do you know if they are home or not? What if they are sleeping or don't respond to knock or doorbell for whatever reason, and shoot you when you come through thinking you are a burgular?
Tripping breakers shouldn't happen very often, if at all. Where is the water shutoff? That would cause more damage. I think you have to live with it or move and maybe if the above are against code, it will give you legal reason to break lease. If the apt. is OK otherwise, I would give it a try and see if it really is a problem.