At what altitude does aerodynamic forces become negligible?
Basically, you are asking the altitude at which the atmospheric forces on the aircraft (lift and drag) become negligible i.e. the point where the density of the atmosphere is so low that the impact of air molecules on the aircraft (or spacecraft as it may be) has little effect.
Generally speaking, as the altitude increases, the density decreases, but there is no fixed altitude above which it disappears- rather it gradually disappears into irrelevance. Even spacecraft hundreds of kilometers above earth's surface in low earth orbits (LEO) have to fire their boosters periodically to overcome atmospheric drag.
The image below shows the height of International Space Station (ISS) over time, which shows the loss of height due to (among other things) atmospheric drag.

This plot shows the orbital height of the ISS over the last year. Clearly visible are the re-boosts which suddenly increase the height, and the gradual decay in between. The height is averaged over one orbit, and the gradual decrease is caused by atmospheric drag; Image from heavens-above.com
Even at that altitude, the drag can be significant for the fuel available and as NASA points out,

For most of the last decade, ... the International Space Station they were circling the globe at an altitude of approximately 220 statute miles, or about 350 kilometers.
... European Space Agency’s resupply ship Johannes Kepler... will bring the fuel needed to boost the station to its normal planned altitude of 248 miles, or 400 kilometers.
“As solar activity rises, the atmospheric density in our altitude range increases causing increased drag on the vehicle. This in turn causes us to have to raise the orbit more often.”
Even though the space station orbits in what most people on Earth would consider to be the “vacuum of space,” there still are enough atmospheric molecules ... that the cumulative effect of these tiny particles contacting its surfaces reduces its speed and causes a minute but continuous lowering of its altitude, or height above the Earth.

The question now becomes different- at what altitude does effects other than drag become much more important. Though we cant set a precise value as above, it is possible to set a theoretical altitude above which the drag can be neglected (not entirely) and other things become more important. This is called the Kármán line, which Von Kármán explains as:

... flew 2000 miles per hour (3,200 km/h) at 126,000 feet (38,500 m), or 24 miles up. At this altitude and speed aerodynamic lift still carries 98 per cent of the weight of the plane, and only two per cent is carried by centrifugal force, or Kepler Force, as the space scientists call it. But at 300,000 feet (91,440 m) or 57 miles up this relationship is reversed because there is no longer any air to contribute lift. Only centrifugal force prevails. This is certainly a physical boundary ...

(Quote from Von Kármán's autobiography, from Wikipedia)
As the drag force is similar to lift (both are proportional to the square of speed and directly proportional to density), it only makes sense to consider that the drag loses its effect on the air/space craft, at least for ease of definition.
From Fédération Aéronautique Internationale:

Astronautics needed the lack of atmosphere to be viable; Aeronautics needed the presence of atmosphere. And atmosphere existed near the Earth’s ground, but did not exist far above the ground. In Astronautics, speeds impossible to maintain in atmospheric drag could be kept for very long periods without power applied to the vehicle.
In Aeronautics, level flying higher and higher meant to deal with less and less dense atmosphere, thus to the need of greater and greater speeds to have the flying machine controllable by aerodynamic forces. A speed so big in fact, that, above a certain altitude, could be close or even bigger than the circular orbital speed at that altitude ...
A lot of calculations were made, and finally it was reached the conclusion, accepted by all scientist involved, that around an altitude of 100 Km. the boundary could be set.
The 100-Km altitude, ever since named the “Karman Line”, came thus into existence as the boundary separating Aeronautics and Astronautics.

Basically, for the purposes of your question, we can take the Kármán line as the point where the atmospheric drag loses its effect on the aircraft, which becomes a spacecraft.