Is it true with JPEG picture files the more times you save them, the more they are compressed?

Depends what you mean by "the more you save them" and "more compressed".
Each time you open and resave a file, there is a loss of quality.
That loss of quality includes detail loss, and less details means that the file is easier to compress (i.e. smaller).
If you save multiple times a file without closing it, it is like if only last save counts.
That's why the JPEG conversion should only be done at the end, after resize/cut operations.

How much do they compress each time?

I edited my message. Actually they do not really compress "more", there is just a loss of info each time, and that depends on the compression ratio chosen. It is hard to give any number.

I'm asking because I did some DVD captures last night, but I didn't have time to crop away the black borders before I saved them. If I open them, take away the borders and save them again, will the quality suffer more, than if I had done the cropping last night and saved them just once?

If you would not have saved them to jpeg if you cropped them directly, yes there is a small quality loss.
For that reason bmp or png is a good choice as temporary format, as it is lossless: I batch-convert my bmps only when I upload the pictures.