There are several options...
You can use a NAS (Network Attached Storage). This can be from a dedicated device like a Synology NAS Station (I personally use this device) that not only has physical drives served across the network, but you can attach USB external drives. I actually have a hub with 3 external drives (for backup and Time Machine) and a USB flash that holds my music collection. I love this setup (bottom left).
You can use a router with NAS functionality. The Apple Airport Extreme (above right) is a good example of a router with this type of functionality.
AirPort Extreme can turn your external USB hard drive into a secure drive you can share across your Wi‑Fi network and access securely over the Internet. All the documents, videos, photos, and other files instantly become available to anyone on the network, Mac and PC users alike. It’s perfect for file sharing, collaborative projects, and more.
Though I am not a huge fan of "putting all your eggs in one basket," this will make do in a pinch.
Wireless USB Flash Disks. SanDisk has a wireless flash drive and I had the opportunity to use one the other day. I have to admit, it was neat bit of kit; it's not as fast as directly connected USB, but pretty neat and convenient.
Share the USB folder from another Mac/PC. By turning on File Sharing, or remote login, you can access the USB by remotely connecting into the machine with the good ports. Under System Preferences just enable file sharing and add the folders (including the
/Volumes/[NameOfUSB] folder) into the "Shared Folders" list.
If you know how to
ssh into your machine, you can use
scp to copy the files. You can get more info on how to do this from this answer
Hopefully, one of these methods will allow you to access the files on your USB.