Get to command line on your ASA and run the following commands and check they look similar to the following:
ASA# sh run boot
boot system disk0:/asa832-k8.bin
ASA#sh run asdm
asdm image disk0:/asdm-633.bin
This shows that the ASA is configured to use compatible versions of the ASA and ASDM images.
Check which versions have loaded:
ASA# sh bootvar
BOOT variable = disk0:/asa832-k8.bin
Current BOOT variable = disk0:/asa832-k8.bin
ASA# sh asdm image
Device Manager image file, disk0:/asdm-633.bin
These should all match what is in your config, but if they don't then you would see problems as the loaded versions of ASA and ASDM have to match.
Now you need to check what files are actually on the ASA:
ASA# sh disk
--#-- --length-- -----date/time------ path
101 15962112 Sep 01 2010 15:21:40 asa832-k8.bin
105 14497692 Sep 01 2010 15:23:32 asdm-633.bin
NB - you will see more files listed than this! You may see multiple version of the ASA and ASDM files, which is ok, but can cause problems if the config isn't quite right or a file is corrupt.
If you don't have the files you expect in place, look at what is there. If there are corresponding versions of ASA and ASDM on there, update the config to use them - this will allow you to access ASDM and you can upgrade via that if necessary. If there are multiple versions, start with the oldest ones first.
Another useful troubleshooting step is to connect a console cable to the ASA, get a console session up, and watch the output when booting - the ASA will report errors during the boot process, but you won't see them if you ssh in. Many errors will result in an ASA that functions, just not as expected.
You could also view the logs, but I prefer the above, as it shows you the errors that occur during boot, as they occur.