Can I keep using a corrupted SD card?

Well, you could. But you might not like the results from doing so.

Based on the rest of your description, though, before you toss the card you probably need to establish that the card, and not something else, is the actual source of the problem.

Here's a checklist of things you should look at:

• How was the card originally formatted? You should always format a new card in the camera in which you are using it. (Note: you can also format the card with a computer as long as you use the correct file system and create the correct folder structure that the camera requires - but it is a lot easier just to use the camera to automatically do all of that.)
• When was the last time the card was reformatted? Not reformatting flash memory cards periodically can lead to file corruption. When a flash memory card is formatted the controller maps out sectors marked as bad so they are not reused.
• Have you confirmed that the card reader is not the issue? Do other cards have similar issues? Or do other cards work fine?
• If the card reader is a portable one, have you tried using a different cable to connect it to the computer? A different port?
• Have you recently updated the USB (or Firewire, or whatever interface you are using) drivers?
• Did you remove the card from the camera or card reader while the card was being written to? This can cause corruption of files. It may or may not have also damaged the hardware of the card.

If you've eliminated each of these possibilities then the chances are pretty good the card was corrupted.