Which Corp and Runner pairs have you been playing with? The suggested Jinteki/Shaper match in the rulebook is pretty poor, because the Jinteki ice in the Core Set is poor and lacks stopping power, and Shaper have a very strong economy option in Magnum Opus.
The Runner is stronger than the Corp in the Core Set. That's partly a slight balance thing, but also because the preconstructed Corp decks are probably worse than the Runner ones. For example, Core NBN has lots of ways to apply tags but only Closed Accounts (and Psychographics) to punish them. On the other hand, Core Weyland has excellent tag punishment in the form of Scorched Earth, but very few ways to apply tags (I think Hunter might be the only way in that deck). There is a clear solution to both of those shortcomings with a little bit of deck building, but that doesn't help with the preconstructed decks.
Another issue is that Corp is, in my opinion, harder to learn to play well than Runner. The Runner has an obvious route to victory while the Corp can feel a little passive. An important concept to learn is that of scoring windows for the Corp, which is those times when the Runner doesn't have enough resources to access. Crudely put, they will pop up when the Runner has just spend a lot of credits on installing programs (etc) or making a run. The Corp player can force this by tricking the Runner into expensive runs, for example by creating a new two ice remote server and playing a card into it. If that card isn't actually an agenda, then maybe you've just tricked the runner into spending credits for no great gain and now they're too poor to get in the next time.
Anyway, my suggestion would be to play with HB as the learning Corp. It's probably the most solid deck in the Core Set and multiple bioroid ice like Ichi 1.0 and Heimdall 1.0 on the same server will be costly for the Runner to get through.
Alternatively, experiment with a little bit of deckbuilding to fix up the weaknesses in the Corp decks. If it's any consolation, in the current meta of every expansion pack released, the Corp is probably slightly ahead of the Runner, so it's not a problem which is inherent to the game system.