To connect aspirator to a threaded faucet :
1) First remove the aerator piece.
2) Buy a metal connector part named "Dual Aerator to Male Hose". I found this at my local hardware store. This provides a very reliable and solid connection to your faucet. If you do this, now you can screw a garden hose to your faucet, right ?
3) Two ways now : connect a piece of tubing like a plastic hose, as short as possible, use metal clamps ( metal rings with screws ) to keep it TIGHT. Aspirator in one side, garden hose to faucet connector in another side.
4) If you can, find screw type connectors till you go down to aspirator size thread, I think it is usually 3/8 ".
Comment on 2 : I was using waterbed plastic connector, very fragile. The metal part is much better. My aspirator connection is now very solid and reliable and does not leak one drop. I can screw this in/out of my faucet when I need it in two minutes. Of course then faucet 100 % back to normal state.