Pokémon GO Jessie and James counters are straightforward. They use un-evolved Pokémon, so you shouldn't have any problems.
To actually battle Jessie and James, look out for their Meowth balloon. You engage them like other Rocket Grunts in balloons.
You will battle Jessie and James one after another. Jessie will use her signature Ekans, followed by Scyther and then Bagon. James, of course, will lead with Koffing. His next two Pokémon are Sneasel and Beldum.
Pokémon GO Jessie and James Counters
Ekans and Koffing are both Poison-type, so any Psychic or Ground-type will make short work of them. As for the others, Scyther and Bagon share a weakness in Ice. James's Sneasel and Beldum are both weak to Fire. Scyther and Sneasel are also both weak to Rock.
Rhyperior is a good choice against all Jessie's Pokémon, as the Rock and Ground typing resist most of their attacks. It can do some work with James, too. Beldum doesn't have any charge moves to fight back against the Rock-type Rhyperior. Even Golem is a solid choice against them.
Bringing any fully evolved Pokémon to the fight is likely overkill since Jessie and James's Pokémon don't actually have any threatening moves. Defeating the duo will earn you 500 Stardust and a Rocket Radar component each.
Jessie and James won't put up much of a fight, so there's an extra bonus to battling them: a chance to catch a Shiny Ekans or Koffing. Defeating them will get you and encounter with Shadow Ekans or Shadow Koffing, and both have a chance to be Shiny.