Backup airspeed indicators (lack thereof)
In order to answer your question first we need to figure out what we're trying to accomplish with airspeed. For the most part the goal is "envelope protection" - keeping the pilots from either stalling the aircraft or overspeeding (and overstressing) it.

So the first thing to know is thaton modern transport category aircraft there are already backup systems to compensate for a failed airspeed indicator.  

For starters there are usually at TWO (visible) airspeed indicators (one for each flight control station - pilot &amp; co-pilot). The airspeed indicators are usually using two different pitot (ram air pressure) sources, so that if one pitot system or indicator fails the aircraft can still be flown on the others.
In addition to those two indicators there is often a third "tie-breaker" system on modern aircraft which is used to determine which system has failed (if two systems agree the third is declared insane and ignored).

In the unlikely event that all of the pitot-based airspeed indicators fail they aren't the only system providing envelope protection: Angle-of-Attack vanes are used to determine when the wing is approaching the critical angle of attack (and is in danger of stalling). These operate independently of the pitot system as a redundant form of stall protection, and as with the pitot system aircraft usually have more than one AoA vane installed.

Both the pitot-based airspeed indicators and the AoA vanes are critical systems, and are heated (to prevent ice from accumulating and interfering with their proper operation) and monitored (by onboard computers and/or the pilots).

Finally in the face of a failure of all of those systems there is the ultimate fallback method that others have described: Pitch + Power = Performance.
A pilot should be able to at the very least maintain level flight, descend, climb, and make standard-rate turns within a safe flight envelope by reference to engine power and aircraft attitude.
While that's far from an ideal situation it is one pilots are trained for.