Scripts are there, but same scripts are not creating any issues on other machines. I mean to say after script completion on other system "Geodatabase is accepting connections" is checked. Only to a particular system it is Unchecked.
This doesn't concern systems. It depends on what is being done using these Python Scripts and which Geodatabase is that pointing to in the script.
I can execute a Script on Server-A, which will modify Geodatabase that is on Server-B. So this is possible.
My suggestion, is to go through all the Scripts, in which this particular Enterprise geodatabase is referenced and check if any of them, is scripted to stop Accepting Connection for that particular gdb.