Safe programming is a subset of correct programming -- every exploitable vulnerability is, basically, a bug such that the consequences of exercising the bug are advantageous to an ill-intentioned individual (the "attacker").
Languages offer facilities which help or not help in making correct programming. For instance, if the task at hand involves doing things with character strings, languages that have an inherent type for character strings, handled as immutable values, make things much easier for the developer. Many modern languages such as C#, Python, PHP or even Javascript fall in that category. Conversely, more low-level languages like C and Assembly require the developer to handle allocation for buffers holding strings, taking care of all lengths and deallocation at the proper time. In that sense, C and Assembly are "less safe" languages than most others when it comes to handling character strings, because they require more care from the developer. "Care" is a scarce resource.
In the same vein, some languages include a lot of "magic" such as automatic trans-typing, which means that a seemingly simple expression may hide a lot of complexity, that may bite the unwary developer. Some modern languages such as C#, Python, PHP or Javascript have that flaw. Conversely, low-level languages such as C and Assembly tend to be "safer" in that everything they do is explicit. The developer just has to read what is in front of his eyes, with less mental inferences on what the interpreter/compiler will do. "Thinking time" is a scarce resource.
There is no paradigm that is inherently less safe than any other; however, lack of a defined paradigm is assuredly less safe. Safe programming, ultimately, is about a developer who knows what he is doing, and this comes from discipline.
Edit: "Discipline" means never writing any code whose complete behaviour, under all possible combinations of normal and abnormal input data, is not fully understood by the developer. In particular, it implies not throwing code together and tweaking until the thing accepts to run. Similarly, copy&pasting from Stack Overflow is poor discipline -- reading solutions on SO is very fine, but it must be followed by thorough understanding of what the code does.