Base Line Road is named for the San Bernardino Base Line, the topographic baseline for Southern California mapping surveys. The San Bernardino city webpage and the Press-Enterprise have some interesting information on San Bernardino’s Base Line Road, including a description of the original monument erected on Mt. San Bernardino to demark the San Bernardino Meridian and Base Line. (You can even hike to that monument, and virtual cache GCM08Y is waiting for you there.) The baseline spans the width of Southern California; that Base Line Road you’ve seen out at Joshua Tree? Same baseline. There are times when the road deviates a bit from the mapping baseline, and sometimes the street name changes to something else for a while, but the baseline namesake is always the same.
At some point in the past, the decision was made to not have Base Line be contiguous where it crosses Foothill. You can make turns onto and off of the eastern section of Base Line, but if you want to drive or park on the section of Base Line west of Foothill, you will need to drive over to San Dimas Canyon Rd and then onto Base Line from there.