Question: How does ATC control traffic without radar?
Answer: In the U.S., the Air Traffic Control system was designed to be operated without radar. Even with the wide availability of ATC radar the current underlying "non-radar" procedural rules and criteria are still an inherent and necessary component of how (primarily) IFR aircraft are routed and, to a lesser extent, separated.
With some noted exceptions (e.g., published "Radar Required" procedures, RNAV routing), Standard Instrument Departures (SIDs), VOR Airways, Standard Terminal Arrivals (STARs) and non-RNAV Instrument Approach Procedures (IAPs) (some exceptions apply) are all examples and part of the non-radar structural design underlying the ATC system.
Although the use of radar and ATC radar procedures are the predominant tools (being replaced by ADS-B), the existing non-radar procedures can be utilized for handling and separating IFR aircraft even if all of the radar systems/antennas were to cease operating. ATC system capacity and efficiency would suffer, but IFR aircraft could still be routed and separated (using non-radar procedures) the same as they were prior to radar becoming available for air traffic control purposes many years ago.
Lastly, normally in today's ATC environment a co-mingling of non-radar and radar procedures for routing and separation is used as necessary depending on the existing circumstances. These circumstances vary during an IFR flight from departure to destination and are usually functionally transparent to the pilot.
Most non-radar ATC procedures are typically more complex to explain and apply than radar procedures. As such, a deeper understanding can be gained by reviewing Chapter 6 of the FAA's JO 7110.65Z (Air Traffic Control Handbook).
Non-radar IFR separation procedures are (primarily) divided into 3 main categories - Longitudinal, Lateral, and Vertical: (Below are samples and do not explain the full use of each category.)
(Refer to the Chapter 6 link above for a full understanding and use of each category)