They are both correct and I don't really think it matters when you use which. :"Attempt to do" is shorter so I would probably stick with that one, it also sounds a bit more natural and concice... unless you have a specific reason for using "attempt at doing". In some specific circumstances it may just sound better or more emphatic but that's not something you can learn with rules as far as I know. I have definitely heard it many times but I doubt there is any situation where it would be incorrect to use "attempt to do something".

My attempt to convince her to stay was unsuccessful.
My attempt at convincing her to stay was unsuccesful.

Both sound fine IMO.

There was an attempt to kill the president.
*There was an attempt at killing the president.

2nd one sounds weird.

I dno. Stick with the first unless the urge to deviate is overwhelming.