I can't speak to the show because I've never seen it but I can talk a little about espionage between the US and Soviets in the 1980s.

First, it's important to realize that it wasn't just the KGB we had to worry about. Like the U.S., they had more than one intelligence organization. The KGB is the most well known, much like the CIA in the U.S., but there was also the GRU, which was the Soviet military's intelligence arm. I won't go into all of the U.S.'s assets because there are 18 different agencies but just be aware that espionage was being conducted by a lot more than just the CIA and KGB.

As to capabilities, yes both were equal. A lot of resources were spent on conducting espionage and on counter intelligence. Both sides were constantly sending false informants to the other's embassy just to waste their time. Also, Aldrich Ames and Robert Hanssen were both active as traitors working for the Soviets during this time, which gave a lot of information to the Soviets. These are the ones we know about. It is also speculated that there was a third spy providing classified info to the Soviets that we never discovered.

Also, both sides know who the intelligence agents from the other side were. The FBI and CIA knew who the Soviet agents were in the U.S. and the KGB and GRU knew who the U.S. intel agents were in the Soviet Union for the most part. This meant that information transfers and meetings had to be conducted very carefully because surveillance was always watching these people. Robert Hanssen never met directly with his Soviet handlers and Aldric Ames only did a few times. Instead a lot of dead drops were used and elaborate surveillance detection routes (SDRs) were used to lose surveillance tails. U.S. agents in Moscow would go (*edit one 2 to 3) on 2-3 hour SDR walks to lose their tails just for a 5-10 minute meeting or quick transfer.

Now influence is a more dicey matter. There are a few things that aid intelligence agents/organizations when they get caught. First, is diplomacy. If you make an example or public spectacle out of your enemy, they will do the same to you. If you kick an intel agent out of your country without catching him red handed then your agents will just get kicked out of their country. A lot of times, the goal of counter intelligence isn't to stop the enemy intel, it's to identify them and stop the people providing them information. Another issue, is if you make a public spectacle, expecially in the U.S. with all of its freedoms, then you risk the classified information being exposed publicly.

Finally, when it comes to convincing people to betray their country or to provide you with help as an intel agent, MICE is the key. Money, Ideology, Coercion (blackmail), Ego. You appeal to one of those four areas. The U.S. loved to use money and ideology during the cold war. The Soviets loved to use coercion and ego (Ames and Hanssen both betrayed their country for ego and money). The Soviets devoted a lot of time and energy into collecting blackmail material against U.S. persons for the purpose of coercion and gaining influence.

Sources:

The Main Enemy: The Inside Story of the CIA's Final Showdown with the KGB by Bearden and Rissen

Spy Handler: Memoir of a KGB Officer - The True Story of the Man Who Recruited Robert Hanssen and Aldrich Ames by Cherkashin

A Century of Spies: Intelligence in the Twentieth Century by Richelson

and if you want to know more about the U.S. Intelligence Community organization and operation:

The US Intelligence Community by Richelson