I have changed the password on my wi-fi and now my Airport base station doesn't not appear in the AirPort Utility. How do I solve this?
If you are saying that the AirPort Express connects to your main router using a wireless connection, and you changed the password on your router.........then the AirPort Express is still programmed to connect to your wireless network using the "old" password. So, the Express is lost and does not know what to do. AirPort Utility will never find the Express unless you connect your iMac to the Express using a wired Ethernet cable connection to change the "old" password to the "new" password.
As Tesserax mentions......and it is water under the bridge at this point......you should have changed the password on the Express first, then changed the password on your router.
If you cannot connect the iMac to the Express using a wired connection, then the simplest solution would be to reset the AirPort Express back to its original default settings using a Hard Reset and then set up the Express again to connect to your network.......just like you did the first time that you originally set up the AirPort Express.
That way, the Express will pick up the correct settings that it needs to connect to your wireless network using the new wireless network password.