This is a typical issue with older HP printers and Windows 7 (I have two HP laserjets). What I did to get them going was to download the newest driver, i.e. XP or Vista and then do a manual install through the control panel (i.e. add a new device or printer) with the printer already hooked up. It will take a bit of time and the install may not complete, but be patient. After a bit, simply restart the computer and try to print.
I read the OP to mean that the problem was not the drivers but the physical connection. An LJ4 only has a parallel port, which most modern PCs lack.
Is wireless networking the only possibility (there are solutions but they are expensive)? If wired networking was acceptable there are cheaper solutions. Basically you want an ethernet adapter for the LJ4, either an external one or one of the HP internal ones (they were quite expensive).
Or, if you want to risk a small amount of money, go for this one on eBay. I've no idea whether it would work with your LJ4, but can't see any reason why it wouldn't.
How do I get Windows to recognize my legacy HP LJ4 which is a workhorse of a printer? I tried a Belden Adapter, but it doesn't work with Win 7 x64. Is there a way to put the printer on wifi?
I'm not sure what I did wrong/right, but this morning I tried once more see if I could get the HP LJ4 to work. On the fourth try to "Windows Update" the manufacturers' box finally showed a listing for HP LJ4. One click and about two minutes of processing had the printer up and listed as my default printer. I don't know why it took so long for the right information to show, but I'm happy it did. I had spent about an hour on the HP home page, Google searches (which led to "give me your money and I"ll tell you") sites without any valid answers to the problem. But it's fixed and I thank those who answered for their efforts.