Up until OS X 10.8 Mountain Lion, you could easily access your eID card reader. Then, suddenly in 10.9 Mavericks, this access was broken. You would try to install the card reader, and the software installation would prompt you to plug in the reader, but your computer would not recognise the card reader.
It looks like these drivers that came with the standard eID software are not compatible with 10.9 Mavericks.
If you take your eID card reader, and flip it upside down, you can find a label. On the label, you can make out the model number and the supplier brand.
Most Belgian eID card readers have a model number such as ACR38U. If you Google on this model number, you reach the website of acr38u.com, which is the site of the eID card reader supplier. If you follow the Drivers link, you can download there a driver for your Mac for OS X 10.9 Mavericks.
Download the driver, and double click on the file name acsccid_installer-1.x.x.dmg. This will mount the disk image. A Finder window opens with acsccid_isntaller.mpkg. Right-click on this file and choose Open.
Once the installation is finished, go back to the eID installer. Run the installer. You will notice the installation will now recognise the eID card reader and your eID card.
One note on performance: When the card reader is plugged in, a process called pcscd might peak at 99% in your Activity Monitor. This will happen while the card is plugged into an USB port, even if there is no eID card present. This can affect battery life and performance of your computer.