AMD Athlon 64 can run 32-bit software just fine.
Do you have any scripts or add-ons, or just a clean mIRC as it comes out of the fresh installation kit?
If yes, remove them. Or, just to see if those scripts are the cause, download the installation kit again from one of the official download mirrors (
http://www.mirc.co.uk/get.html), install in a different folder, run from there and see if it continues shutting down like that.
If not, go on reading here.
Do you have Windows XP with Service Pack 2?
If yes: AMD Athlon64 and most recent Sempron processors have a "Data Execution Prevention" or "NX" (No Execute) flag which basically will stop running a piece of software in certain situations - this was implemented as a measure to detect and stop dangerous code (such as viruses). Windows XP with Service Pack 2 can disable or enable this feature. It is preferred to have it enabled to gain some extra performance, but for the legitimate application that are having problems you can add them to an exception list.
I am guessing that this DEP setting could be the cause for mIRC shutting down. Give it a try then see if it helped. Go in Control Panel > System > Advanced > Performance Settings > Data Execution Prevention tab, select "Turn on DEP for all programs and services except those I select", then click the "Add..." button to add mIRC to the exception list. (You will probably have to select the mirc.exe file, I don't have Windows XP so I can't tell you what else you'll need; just follow the options and use the Help buttons.)
Let us know if this guess fixed your problem.
