They still used film back in 1961, and West Side Story has no space aliens or robots to put in with either CG or stop-motion. There are a few visual tricks present, mostly based on optical illusions (such as the moment at the dance when Tony and Maria first meet). For the rest of it, however, what you see is what you get. The magic comes in the Jerome Robbins choreography, not any tricks with effect or computers.
It's actually impressive that they used the actual New York locations for shooting much of the time. Musicals in that era almost never took place outside of a sound stage. Why? There's a lot of people moving around in those big dance numbers, and getting it right requires very controlled conditions. If you put it out in the real world, you lose that control. Clouds and rain happen. Pigeons poop. And while the street is likely closed off, that won't stop some random person from inadvertently wandering through the shot and pointing at the nice people dancing.
In fact, during filming on the streets, the cast was harassed by some of the locals, who threw rocks and dropped stuff off roofs—to the point that the producers had to hire real gangbangers to protect them. (Source)
Director Robert Wise had to convince Robbins that filming the Prologue on the mean streets would be possible:
Jerry agreed with this, but he said, 'You've given me the most difficult task right off the bat: to take my most stylized dancing in the piece and put it against the most real backgrounds we have in the picture.' He struggled with it. We made tests in downtown Los Angeles streets in daylight. We had a rig running around the studio streets with Betty Wahlberg [rehearsal pianist] at a little piano on a trolley and an umbrella over her. She'd be pulled along as she played and the dancers would rehearse along the streets as Jerry studied, developed, and adapted the dance steps to the outdoors and the sunlight. (Source)
After the "Something's Coming" number, the rest of the action takes place in the evening or at night. Wise thought that rooftops, alleys, the gym, etc., could be convincingly constructed on a soundstage. But the opening prologue, dancing through the West Side, sets an authentic tone that carries through the film.