Salzburg is small enough to cross on foot in half an hour, so the bases differ less by convenience than by noise, price and which side of the Salzach you wake up on.
Altstadt left bank — Getreidegasse and the fortress
The UNESCO core beneath the Hohensalzburg, with the Getreidegasse, the cathedral squares and the Festival halls. Everything is on the doorstep and so are the tour groups; rooms are small, old and dear, and delivery vans start early in the narrow lanes.
Right bank — Linzergasse, Mirabell and the Kapuzinerberg
Across the Salzach, five minutes' walk from the old town but a different price bracket. Mirabell gardens, the Linzergasse shops and the steep Kapuzinerberg path are all here, and the main station is walkable, which matters if you are day-tripping by train.
Nonntal and Riedenburg — the quiet edges
Residential streets tucked behind the Mönchsberg and under the fortress's southern side. Ten to fifteen minutes on foot into the centre, noticeably cheaper, and you get the Mönchsberg paths as your local park rather than a queue.
Practical: the Festival changes the arithmetic
From late July through August the city is full and prices roughly follow. If you are caught out, the S-Bahn line south through Hallein and Golling puts you within half an hour of the centre for much less — but note that the Salzburg Card covers city public transport and specifically does not cover the S-Bahn, so budget for those fares separately.