Kusatsu Onsen is not provided with a railway station, so if you want to reach it with mass transportation, you have to take a bus from the closest stations, which are Naganohara Kusatsuguchi (Agatsuma Line) or the more known Karuizawa (along the Hokuriku Shinkansen).
From Naganohara Kusatsuguchi Station
If you are heading from Tōkyō, you are likely to reach first Takasaki Station, the main access terminal of Gunma Prefecture. If you ride a Shinkansen, in less than a hour you will be able to reach it, or in a couple of hours if you use a local train (take the Takasaki Line from Tōkyō or Ueno Station, or the Shōnan-Shinjuku Line from Shibuya, Shinjuku and Ikebukuro).
Once in Takasaki, transfer to the Agatsuma Line, to getin about 80 minutes to Naganohara Kusatsuguchi Station. From here, you can find a connecting bus to Kusatsu Onsen Bus Terminal (about 30 minutes).
From Tokyo to Karuizawa by Shinkansen is a quick and comfy 1 hour journey. However, from here you need to ride another bus for about 80 to 150 minutes (according to the route, as the one passing close to Mount Shirane is way longer, but pretty views are assured), to reach Kusatsu Onsen. These links are operated by Kusatsu-Kōtsū and Seibu.
With expressway buses
You can also choose a slower, but transfer-less bus connection from Tokyo to Kusatsu. There are different companies offering this, and it is going to take at least 3,5 to 4 hours one way.
Maybe you are not likely to rent a car in Tokyo to reach all the way Kusatsu, but if you are roaming on the road in the northern part of Japan you will be able to reach Kusatsu by getting out at Shibukawa-Ikaho IC from the Kan’etsu Highway (it takes about 2 hours from Takasaki).