There is lots of info here I will go back a bit further before the 60's for a complete answer. Dedicated flight dispatchers date back as far as 1936 when they were broken off from ATC duties into their own functionality.
Weather planning involving ground based observations and intercommunication dates back as early as the 1930's when the technology made inter-airport communications possible and subsequently en-route weather planning possible before takeoff.
In 1930, Boeing Air Transport installed the Western Electric
radiotelephone system in its aircraft and 18 ground stations. Ad copy
for Western Electric Aviation Communication Systems referred to Boeing
Air Transport pilots as “always in touch with dispatchers and weather
observers along their routes. Reports on weather and field conditions,
guiding radio beacon signals, or instructions come in clearly, helping
pilots to bring their ships through on time”
Formal dispatch training goes back to at least 1936 and had various requirements. There is some great footage from a 1947 film about the "flight superintendent" here (at 4:55) you should watch.
The first documented educational program for training aircraft
dispatchers appeared in the October 1936 issue of Aviation. The
subjects covered included meteorology, dispatch practice, and airline
operations. The course took 18 months to complete and required 2 years
of college or 9 months of accredited engineering college coursework
prior to enrolling (p. 66). The 1947 vocational film Your Life Work
Series: Air Transportation called the dispatcher a “flight
superintendent.” The film describes a flight superintendent’s key role
in all flight operations: "He is the man who decides whether the
planes will fly or not. He releases all planes on his division,
follows their progress in the air, and keeps the captains…advised of
conditions affecting their flight…He coordinates all flight operations
to achieve these objectives: safe, swift, and dependable air
transportation." (Twogood, 1947) The flight superintendent position
required a pilot certificate and a dispatcher certificate issued by
the Civil Aeronautics Authority (Twogood, 1947).
En-route pilot reports (PIREPS) date back to Feb 1 1943 so pilots would have had access to real time conditions over the radio from ground based stations going back well before the 60's allowing pilots to make en-route weather choices and ground dispatches to make pre-flight choices.
Feb 1, 1943: CAA inaugurated an expanded flight advisory service at
all air route traffic control centers. The centers originated
advisories on weather changes and hazardous conditions, and airway
communication stations relayed this information to nonscheduled
pilots. The service provided these pilots with some of the assistance
that airline pilots received from their dispatchers. In Jul 1943,
CAA's communication stations also began a flight communications
service. When contacting pilots by radio, communicators were
instructed to volunteer information on important weather changes or
inoperative facilities along their route.
This copy of Aviation Weather dates to 1975 (and is pretty neat) but the preface indicates that its publication goes back as far as '43 indicating dedicated weather information was published going back at least that far (a copy can be had if you really want).
Thunderstorm avoidance was (and still is) a big issue. A good set of MK-1 Eyeballs is always your first option but on board radar tests date back as early as 1949 with the first units coming around a decade later Which would allow pilots to offer far more accurate PIREPS
Since the advent of commercial aviation, thunderstorms caused delays,
diversions, and accidents. Airlines attempted to find ways to avoid
enroute weather while maintaining scheduled operations. In 1949,
American Airlines and the Unites States Navy conducted tests of an
onboard weather radar system installed in a Convair airliner (“Flying
Lab,” 1949, p. 28). In 1956, American Airlines equipped its fleet of
DC-7s with airborne weather radar (“AA to Equip,” 1955, p. 20).
Airlines quickly embraced the innovative technology, and Aviation Week
(Christian, 1955) quoted a Northwest Airlines pilot: “We all wonder
how we ever got along without it. Pretty soon the public won’t fly in
anything but radar-equipped airplanes” (Christian, 1955, p. 40).
The tricky part becomes trans-oceanic flights where there may not be any reporting stations. The importance of trans ocean (Atlantic) weather reporting became important during WWII and the use of light/weather ships became apparent to gain such information. Transatlantic flights would not have been in the dark about weather on account of that.
Here is a pretty neat article about the radios they used to actually make the reports from ground to the aircrafts.
As noted in the comments many airplanes of the era had a Flight Navigator on board, they would have provided a great deal of fuel and navigational planning as that was their job. The FAA still has a handbook covering the tasks of the navigator you can find here.
On any note: Numerical weather forecasting dates back to the turn of the century and by 1950 the national weather service was providing future outlooks forecasts. Paired with airport based reports, PIREPS, and onboard radar systems for more in flight information; Trained dispatchers were capable of making educated decisions on flight dispatching. In some cases these dispatchers were full time employees of the airlines and much like in todays era dispatchers would have been responsible for weather planning and likely en-route planning as well as fuel requirements with final checks landing on the plate of the flight navigator.
