I found the answer myself and I am sharing it here in case someone else happens to need it, too. I found it by performing a simple google search for office+2007+draw+borders :) It was on the "Deprecated features for Excel 2007: Today Eric Patterson returns with a few more guest posts on Compatibility" blog.

The solution is: Ribbon > Excel Options > Customization.

There one should select "Commands not in the Ribbon" from the left drop-down menu. "Draw Borders" is on the list.