The easy answer is that there is not a way to clip the basemap layer to your area of interest, however you can set your map up to only show the basemap for your area of interest even if you don't have ArcInfo or any other licenses besides desktop. To do so, you're going to need a polygon layer with at least two attributes to make a large "masking" poly and a poly that shows your area of interest. For example, let's say I wanted to display a basemap for a single county within a state. I load in my shapefile that shows all of the counties for my state of interest, and then I set up the symbology to create the mask. To do this, right click the shapefile and select Properties, and then select the Symbology tab. Click on Add values at the bottom of the dialog box, and select the county you are interested in, and click Add. Now there are a couple of different rows in the Symbology dialog box - one row for "all other values" and one row for your county. Make sure the checkbox next to "all other values" is checked, and double click the symbol for that value (if nothing happens when you click on it you can right click and select Properties for All Symbols). In the Symbol Selector dialog box choose white as your fill color and also as your outline color, and click OK. Now double click the symbol next to the county that you added (or right click and select Properties for Selected Symbol(s)). Change both the fill color and the outline color to "none." Click apply, and you'll be left with a polygon where the area of interest is hollow and the surrounding area(s) are masked. You can add your basemap underneath, and only your area of interest will be shown.
I hope this helps!
