If you have ffprobe installed:
ffprobe <input> -show_entries format=nb_streams -v 0 -of compact=p=0:nk=1
You can also filter for video or audio streams by adding -select_streams v or -select_streams a, respectively. See the manual for more details.
If you do not have ffprobe, ffmpeg can be used too, but in a slightly less robust way (i.e., this may break if the output format changes or something contains the words "Stream #" in the output):
ffmpeg -i <input> 2>&1 | grep "Stream #" | wc -l