Bring up your Settings pane and select Applications. In the Applications screen, select USB Settings and select Mass Storage. Now, when you connect the handset to your computer, pull down your notification bar and select USB Connected, it will bring up a prompt about mounting the volumes - press Mount. In the future you will just need to plug it in and mount the drives from the pull down menu.
You should be good to go. If not, try installing the Captivate drivers.
Also several people have had issues with plugging the Captivate into the front USB ports on their computers, and switching to a rear USB port fixed it - so give that try if you are still having issues.