Are you able to use the screen of the old Mac at all? Is it totally disabled or is it in a semi-usable cracked/distorted state?
If you
can use it enough to see what you're doing, I would suggest following the steps
here to open Migration Assistant on your old Mac and on your new one and then connect them. Pretty sure that if you just connect them without that step, it won't do anything.
If you can't see the display of the old Mac at all, I think your best bet would be to connect it to an external display of some kind temporarily so you can start it up, open Migration Assistant and transfer your data (and perhaps wipe the old Air clean before you get rid of it).
If you don't have a display available to borrow, perhaps the Apple Store would let you use one for a bit if you brought both machines in. A local computer shop of some kind might also be cool about it and let you plug one of theirs in. And I guess as a last resort you could buy a basic portable display (like
this), use it, and then return it when you're done. Gray area, ethically, I guess, but people do stuff like this all the time.
By the way, also be sure the cable you're using supports data transfer. Some do not, and are only for charging. Possible if you're doing this at the Apple Store they'd let you borrow a fast data cable if you ask nicely
