Yes, you can use AmigaExplorer with TCP/IP. To do this, you'll need a Network Interface Card, and a TCP/IP stack.
I forget, but I think you had the A600. If so, your best bet is an old version of AmiTCP (demo available on Aminet). I use a Kingston KNEPC2T PCMCIA NIC with Harry's CNET.DEVICE. You can also get a driver for some 3Com cards.
It'll be a bit faster than serial, but still rather slow.
If you buy a NIC, make sure to get a 16bit PCMCIA not 32bit CardBus.