Just to clarify the situation. You are trying to send files from the computer to the iPhone? You can see the iPhone as an option, but nothing happens when you select it?
Have you tried to transfer an item from the iPhone to the computer to see if things work in the other direction?
Try turning Air Drop off on both devices, then turning Air Drop back on for both devices.
Are you running any anti-virus apps, cleaning/optimizer apps, or third party security software on either the computer or the iPhone? If so, then uninstall those apps by following the developer's instructions since those types of apps are not needed and usually causes more problems than they solve.
FYI, every time I have had an issue with Air Drop, a reboot of both devices and/or updating the OS on both devices has solved the problem. Double check that both the iPhone and computer are completely up to date for their OS patches. Keep in mind that iOS 16.5 was just released as mentioned by @leroydouglas. AFAIK the current version for macOS Ventura is 13.4.
If you are using an older iPhone running an older version of iOS, then maybe Air Drop is no longer supported. We have been assuming you are running iOS 16.5 on the iPhone and macOS Ventura 13.4 on the computer. If these assumptions are incorrect, then please let us know.