Assuming that projectile weapons - javelins, bows and arrows, etc. - were used somewhat commonly during gladiator matches, you would think there's a decent chance that spectators would be harmed. Do we have any sense of whether that was the case?
They did, but not from arrows or spears.
Although the amphitheaters and converted theaters used for gladiatorial combats varied in design, most were set up more or less like the Colosseum, where the arena was separated from the first rows by a tall masonry wall crowned with netting. These barriers were sometimes quite elaborate; in the temporary wooden arena constructed by Nero, for example, the netting consisted of gold wire laced with amber, and was suspended from hundreds of elephant tusks. The Colosseum's barrier was probably even more impressive (no ancient author, alas, took the time to describe it). If we include the netting, it was about 15 feet tall, and studded with stations for the sharpshooters who monitored the arena (and crowd) for any funny business. These measures sufficed to keep tigers and elephants, let alone gladiators, at bay.
Most gladiatorial weapons, moreover, were designed for close combat. Although some varieties were equipped with spears, these were generally for stabbing, not throwing. No gladiator who wanted to live - and most gladiators, remember, hoped for freedom and/or a handsome reward for their time in the arena - would dare risk his weapon on a throw wild enough to end up in the stands. We might imagine some condemned prisoner with nothing to lose trying to attack the emperor or something, but the sharpshooters would take care of anyone who looked like he was going off-script.
There was, in short, little risk to the spectators during a gladiatorial match, and there is no evidence that anyone in the stands was ever injured by a stray arrow or spear. That does not mean, however, that they were safe. If the arena was temporary, there was a risk that the stands would collapse, something that actually happened in the reign of Tiberius:
"One Atilius, of the freedman class, having undertaken to build an amphitheatre at Fidenae for the exhibition of a show of gladiators, failed to lay a solid foundation to frame the wooden superstructure with beams of sufficient strength....The building was densely crowded; then came a violent shock, as it fell inwards or spread outwards, precipitating and burying an immense multitude which was intently gazing on the show or standing round." (Tac., Ann. 4.62)
Supposedly, no fewer than 20,000 spectators were killed.
If you happened to be in Rome, and if the emperor happened to be a sadist, there was always a slight risk that you would somehow displease him, and end up part of the show. According to Suetonius, the emperor Domitian once overheard a spectator in the Colosseum loudly complaining that the games were fixed. Enraged, he sent his guards after the man, who was dragged onto the arena and torn by pieces by a pack of ravenous dogs (Dom. 10.1). On another occasion, Domitian forced a senator he disliked to fight a lion. More than a century later, the unbalanced emperor Commodus, who fancied himself a mighty hunter, paused in the middle of an ostrich-killing spree to threaten a group of senators in the front rows with a severed ostrich head (as one does).
So the gladiatorial games weren't exactly safe - but you were more likely to be killed by an erratic emperor than a stray arrow.
For those interested in the topic, I go into greater depth about the gladiatorial games and the Colosseum on this page.