Absolutely. Most of the crossings one hears about took place at the Wall because it was actually the *easiest* part of the border to escape over.
The first few years of East Germany's somewhat brief existence were marked by simple border protections, featuring little more than barbed wire fences. This, however, was updated in the 1960s, and the following focuses on the border from then on.
The actual border between East and West Germany (called the "Inner German Border" was *much* more heavily fortified. Less like the Berlin Wall and more like the Korean DMZ. The Berlin border was, naturally, easily visible, and, after the very public shooting death of Peter Fechter in August 1962, East German border forces were ordered not to fire on anybody in public view. This did not apply to much of the Inner German Border. To prevent crossing, there was a 5-kilometer "exclusion zone" before the actual border which was illegal for most East Germans (I believe there were a few exceptions, mostly farmers and such who had occupied that land before the border was established) to enter.
Assuming one made it to the border itself undetected, they would then have to contend with the "control strip," a 1-kilometer section with things such as observation bunkers, guard towers, dogs, and electrified, alarmed barbed wire. Tank traps and ditches were installed to make vehicle crossings more difficult. Several crossings, both successful and unsuccessful, took place using stolen armored vehicles or bulldozers, prompting East Germany to arm border troops (Grenztruppen) with anti-tank weapons.
If one somehow managed to make it through this part as well, the next obstacle was the actual border fence. Some of it was made up of normal barbed wire, but some of the border featured special fencing that was difficult to climb, razor sharp, and impossible to cut through. Imagine a twelve-foot chain-link fence, but made of knives. Around this, there would often be land mines*. Some areas, especially those near villages, had more traditional Berlin Wall-esque concrete barriers.
And if this wasn't enough, the border fence was moved inwards a bit in the 1960s, so that border troops could shoot potential escapees as they ran, without risking a cross-border incident.
This doesn't necessarily mean it was *impossible* of course. Werner Weinhold, for example, was one of the Grenzers assigned to the border, and shot two other Grenztruppen soldiers in his escape. One of the most incredible is the story of Michael Gartenschläger, who in 1971 escaped with five other people, and even returned later to capture an SM-70 anti-personnel mine* for West German intelligence.
But what about the sea? Shore-ly that was an option?
Some people escaped by sea as well. But the Baltic is not especially hospitable. It's cold, prone to storms, and both the sea and the beaches were heavily patrolled by the Grenzbrigade Küste, the naval wing of the Grenztruppen. Again, it wasn't impossible, and several people successfully sailed, or even swam, to freedom in the West.
And then of course there was flight! A few people made it over in small planes, and one group even escaped in a hot-air balloon! Trains were also an option; shortly before the tracks between East and West Berlin were dismantled, a train operator filled his train with escapees and ran it right through the border into the West, in one of the largest and most dramatic escapes of the border's history.
Unfortunately, while many of these attempts were successful, they represent a small fraction, and most weren't. Hundreds of people lost their lives trying to cross. Tragically, the last death at the Berlin Wall was just months before it fell. Ironically, being captured was one of the surest means of escape, as East Germany would often "sell" would-be escapees to the West in exchange for much-needed western currency.
*The SM-70 was the terrifyingly lethal combination of a land mine and a shotgun. These devices were attached to the border fence, and were set off by a tripwire. Once they became known to the West, diplomatic pressure forced East Germany to remove them and replace them with more conventional mines. Here's a diagram I stole: https://www.sciencesource.com/Doc/SCS/Media/TR1_WATERMARKED/f/b/8/5/SS2674212.jpg?d63644009823
Sources available upon request.
