Three countries that neighbor Syria are taking on refugees in mass numbers.
These are Lebanon, Jordan, and Turkey. Iraq would be a fourth, but Syrians are not going there because of the ongoing conflict between Muslims. Turkey, Jordan, and Lebanon are secure enough right now to attract desperate refugees, but Iraq is not.
So, these three neighboring countries cannot adequately stop the many thousands that cross the borders daily. Human compassion also kicks in. Meaning as a soldier is guarding the border, possibly seeing some of the fighting going on, hearing gunfire and bombs, seeing smoke, etc. Well, how can they possibly look at a family carrying children, people with nothing but the clothes they are wearing, and so on, and turn them away back into almost certain death?
They typically will not do it. The Arabic/Middle East culture hospitality also kicks in, and suffering refugees are allowed to cross the border on foot.
Other refugees are crossing by sea, as you hear in the news, too often to tragic ends. The ones that cross by sea are ending up in Europe. Again, same scenario as the three neighboring countries where they are crossing by foot: How can a person turn them away after they are barely alive, assuming they have survived the sea crossing, when they land on shore i.e.:Greece, Italy
They typically do not. Once these refugees land on a European coast, they are put into refugee camps. Sometimes they wait there for years in miserable conditions. So, as you have been hearing in the news lately, some have begun desperate walks from as far south as Italy/Greece, across Europe, to get to Northern Europe, where they hear nations like Germany are welcoming Syrian refugees.
To most nations of the world, including the rest of the Arab countries, Syrian refugees cannot make it there on foot (too much desert), or no other way other than planes to get there (USA, etc.). So, these nations have to do a lot in order to fly refugees in; these nations choose not to fly them in. In the West, their voting tax payers may revolt at such an expense from their governments.
For rich Arab countries expense is not the problem…but rather Islamic extremism, and sect disagreements. For example, Sunni nations do not want to let in fleeing Shiites, Druze, or Alawites, and vice versa. That type of variety would make it much more difficult for the governments to control extremism, or maintain the current state of acceptable peace. So, they do not allow refugees to enter in large numbers, and Western powers do not expect them to nor do they pressure them into it.