Yes, they did. Before I dive deeper let me just clarify that it was not the Soviets who were responsible for securing the border of East Germany, but the Ministry of National Defense of the GDR. In fact the so called "Grenztruppen" (border troops) were an independent branch within the military of East Germany.

The German Wikipedia has a (non-exhaustive) list of known tunnels. As you can see most of these were built between 1961 and 1964 and were collaborations of people in the East wanting to get out and loosely organized groups of so called "Fluchthelfer" (escape helpers) in the West (the latter usually dug the tunnels). As you can see from the fifth column there were several individuals who helped build more than one tunnel. They were indeed semi-professionals at what they were doing and some of them financed themselves through secret funds from the West German government and sometimes by selling film rights. So called "Tunnel 57" (57 for the numbers of refugees escaping through it), for example, was built this way.

How did the East German government react? For one, the Ministry for State Security (Stasi) was directing considerable resources to subvert these groups on both sides of the wall. Aforementioned "Tunnel 57" was discovered one night after it started operating on October 3rd 1964 (what a coincidence!), when a confidential informant gave notice to the Stasi. Other measures consisted in declaring the immediate area close to the border to be restricted and demolishing buildings adjacent to West Berlin (this was also done to realize the final version of the wall in 1975 which required at least 30m of cleared depth). For illustration compare the famous Bernauer Str. in 1961 and 1980. The facades of the houses in that street were the actual border to West Berlin and on August 13th 1961 people could simply jump out of the window to flee. Other counter measures included installing listening devices to detect tunneling projects and the Stasi building their own tunnels along the border, expecting to intersect and discover existing or planned tunnels used by refugees.

TL;DR: Yes, there were tunnels, but there was no extensive network as such, but isolated attempts which were usually quickly discovered. With further development of the wall and an increasingly pervasive state security apparatus it became more and more difficult to build these tunnels.