How do you make crossing wires not connect in EAGLE?
Common practice is to use a two-layer board with tracks (wires) on both sides.  Vias (small plated holes) are used to connect the tracks on the top side to those on the bottom side.  

If you get the boards made by a commercial board shop the holes will normally be plated to provide connections between the sides.  If you make the boards yourself, you'll have to solder wires in the via holes to make the connection.

If you only have a single-sided board (tracks on only one side), the wire jumpers you suggest could be used - many commercial products have single-sided boards with wire jumpers.  You may have to route the board manually to minimize the number of jumpers. 

EDIT - Additional suggestion:

Even if you are making a single-sided board, lay it out in Eagle as if it was to be a double-sided board, then when building the board, replace the tracks shown on the non-copper side with your wire jumpers.  This will allow Eagle's Design Rule Check (DRC) function to check your board layout against the schematic, and tell you of any missed or incorrect connections.

The "air wires" you mention are the lines that Eagle PCB uses to show the required connections.