How do sound cards cope with negative voltage?
Computer power supplies do have multiple voltages, including negative ones, so if some amplifier needs  dual rails, these are available. 

A single-supply pre-amp input is biased to some positive middle-of-the-range voltage, and the input signal is applied to that node via a coupling capacitor which passes AC, while blocking the DC component, preventing that input's bias voltage from driving DC through the source.  (Additionally, a DC return resistor is often provided on the outside of that capacitor in case the output stage of the previous device needs its load to be a complete DC circuit.)

If the input line goes right to some op-amp chip without a coupling capacitor, that chip must be handling the negative voltage somehow. If there is no negative power rail, then the circuit topology is doing it. The Texas Instruments document Single-supply Op-amp Design has a lot of information about this. Look at the Boundary Conditions paragraph right on the first page:


  Use of a single-supply limits the output voltage range to the
  positive supply voltage. This limitation precludes negative
  output voltages when the circuit has a positive supply voltage,
  but it does not preclude negative input voltages. As
  long as the voltage on the op amp input leads does not
  become negative, the circuit can handle negative input
  voltages. [Emphasis mine]


It is possible to design a very simple input stage with a single discrete component that handles voltages more negative than the negative supply to the device. The obvious example are JFET transistors (and, similarly, depletion mode MOSFETS). The pinchoff gate voltage of a n-channel JFET is below the source voltage. You can bias a JFET input at zero, and it will nevertheless handle negative swings without clipping them off. Look this very simple pre-amp for a passive pickup in an electric guitar. Note how there is only a single voltage supply, and yet the pickup, which generates both positive and negative voltages, is coupled directly to the gate of the JFET. Only the output is capacitively coupled.