In the example that you gave, "problem with" would be the correct choice, because you are talking about the cause of a problem. If there is a thing that is causing a problem for you, then you have a "problem with" that thing.
There are other contexts where "problem from" is correct, though. For example, if you were talking to a friend about a math problem in a book, you might say:
Did you finish that difficult problem from Chapter 2?
In this case you use "from" because you are talking about where you encountered the math problem.
Another time you might see "problem from" would be if you were talking about a problem that spanned a certain time period, and were talking about when it began. For example:
The leaky roof was only a problem from November through March,
during the rainy season.
Here, you say "from" to indicate when the problem begins.