On the face of it, this is not a particularly dangerous situation as opposed to could it would be. Turkey's primary purpose for scrambling its jets is to save face, not to direct protect its borders from any hostile action. One of its fighter jets was shot down and it doesn't want to look weak to Syria, the rest of the world, and especially its people. Therefore it has to show an increase in vigilance.
Syria on the other hand is operation on and around the Turkish-Syrian border because, like in Afghanistan, insurgents are using the border as a safe zone where they cannot be engaged by Syrian forces without a costly diplomatic price. However unlike Pakistan and the United States, Turkey is capable of actually effectively engaging foreign forces that cross its border and may do so to save face, or in the act of self defense. Syria has no interest in attacking Turkey which is why I believe the shoot down of the fighter jet was a real accident.
What Syria does have an interest in is attacking insurgents, which again often cross the border seeking protection in the hopes that Syria won't take the risk if the reward isn't great enough and cross an international border with their military. Back when Saddam was fighting the Kurds in Northern Iraq, his troops would frequently cross the border into Turkey however it was done in small numbers, was a brief stay, didn't go far into Turkey, and they were attacking Kurds which are also the Turk's enemy more or less. As a result, Turkey didn't care so much. The Syrian insurgents are not enemies of Turkey directly like the Kurds were, although they aren't exactly friends either because they strain local resources as well as bring fighting with them, however their cause its not directly opposed to Turkey's national interests which means that while they are probably a bother for Turkey they aren't as big of a problem as the Kurds are.
What this means for Syria is they have to tread even lighter when crossing a border, the Turks will be less forgiving for border crossing than they were against the Iraqis for killing Kurds. And again since Syria has no interest in attacking Turkey, we must assume that they are at, on, and over the border to hunt insurgents, and if they want to have Turkey look the other way for the most part they are going to have to play it very very very carefully since the Turks don't share an enemy here with the Syrians like they did with the Iraqis. Therefore I don't think Syria is going to intentionally start trouble with Turkey by undertaking cross border missions that may kill Turkish civilians or destroy Turkish infrastructure, and of course Turkey has no interest in attacking Syria so as long as the Syrians are cautious, which is there only way to hopefully be ignored in future cross border insurgent hunts, there shouldn't be any Turkish/Syrian fighting.
HOWEVER, accidents can happen whether Syrian copter shoots a rocket across the border that blows up a Turkish home thought to hold Syrian insurgents, or a Turkish aircraft spies a Syrian copter getting too close and shoots it down. Or a number of other things that can happen to worsen this situation.
But we can be fairly certain there is no intent on either side to commit violence against the other in an aggressive manner.