It's probably true that C++'s new is safer than malloc(), but that doesn't automatically make malloc() more unsafe than it was before. Did your friend say why he considers it insecure?
However, here's a few things you should pay attention to:
1) With C++, you do need to be careful when you use malloc()/free() and new/delete side-by-side in the same program. This is possible and permissible, but everything that was allocated with malloc() must be freed with free(), and not with delete. Similarly, everything that was allocated with new must be freed with delete, and never with free(). (This logic goes even further: If you allocate an array with new[], you must free it with delete[], and not just with delete.) Always use corresponding counterparts for allocation and deallocation, per object.
int* ni = new int;
free(ni); // ERROR: don't do this!
delete ni; // OK
int* mi = (int*)malloc(sizeof(int));
delete mi; // ERROR!
free(mi); // OK
2) malloc() and new (speaking again of C++) don't do exactly the same thing. malloc() just gives you a chunk of memory to use; new will additionally call a contructor (if available). Similarly, delete will call a destructor (if available), while free() won't. This could lead to problems, such as incorrectly initialized objects (because the constructor wasn' called) or un-freed resources (because the destructor wasn't called).
3) C++'s new also takes care of allocating the right amount of memory for the type specified, while you need to calculate this yourself with malloc():
int *ni = new int;
int *mi = (int*)malloc(sizeof(int)); // required amount of memory must be
// explicitly specified!
// (in some situations, you can make this
// a little safer against code changes by
// writing sizeof(*mi) instead.)
Conclusion:
In C++, new/delete should be preferred over malloc()/free() where possible. (In C, new/delete is not available, so the choice would be obvious there.)