Hello Lee Ann,
It sounds like you may have a bad USB cable, can you try using a new cable and possibly a different USB port on the computer. Might be worthwhile to blow some compressed air into the port on your iPhone if the issue persists after swapping for another cable.
Please be sure you are running the latest version of IOS on your phone, alternatively you could use and iCloud or Google account to upload your photos to the cloud then you could save them to your computer from the either of the vendors online storage.
One of these solutions will ensure you have a safe backup of your photos, let us know how it goes.