You’re right, there’s a big move to online submissions in the world of literary journals. This seems to be a result of several factors: a growing desire to preserve the environment, the efficiency of new submission programs, and a desire to connect with more geographically distant editorial boards and writers. Some journals are even requiring online submissions and no longer accept traditional mail submissions except in special circumstances. For writers who have been licking and stamping envelopes for eons, this can seem unsettling.
Many writers and editors have been pleased with online systems, noting better tracking of submissions, easy notification systems, and shorter response times because of the efficiency of getting submissions to readers. (The Virginia Quarterly Review even noted on its blog an increase in angry letters from writers who were concerned that their response time was too quick.) Cutting out the postal process also means saving some money and frustration: no need to buy envelopes and stamps or worry about upcoming increases in postage for those SASEs.
No system is fool proof. But that’s true both online and with paper. The submission process itself is one founded on trust—that your work will be handled and read with care. The best way to put your mind at ease hasn’t changed: submit to reputable journals.