The accepted answer is still working but outdated since 2013. 
This answer is based and new functions from CMake v2.8.12, v3.3 and v3.13.
Since CMake-2.8.12 (2013)
Two new commands to set CMAKE_CXX_FLAGS:
The documentation of last version has not changed a lot since cmake-2.8.12:
In you case you can use:
target_compile_options(${TARGET} PRIVATE ${BUILD_FLAGS})
Or simply if you have a single target:
add_compile_options(${BUILD_FLAGS})
More examples
target_compile_options(mylib PRIVATE   -O2) # only internal
target_compile_options(mylib INTERFACE -gl) # only external
target_compile_options(mylib PUBLIC    -g)  # same as PRIVATE + INTERFACE
# multiple targets and flags
target_compile_options(mylib1 mylib2 PRIVATE -Wall -Wextra)
target_compile_options(    mylib PUBLIC -DUSEXX)  # Bad
target_compile_definitions(mylib PUBLIC -DUSEXX)  # OK
add_compile_options(-Wall -Wextra) # for all targets in current directory
add_compile_options(-DUSEXX)       # Bad
add_definitions(-DUSEXX)           # OK
Deprecated COMPILE_FLAGS
cmake-3.0 documentation flags COMPILE_FLAGS as deprecated:
COMPILE_FLAGS
Additional flags to use when compiling this target’s sources.
The COMPILE_FLAGS property sets additional compiler flags used to
  build sources within the target. Use COMPILE_DEFINITIONS to pass
  additional preprocessor definitions.
This property is deprecated. Use the COMPILE_OPTIONS property or the
  target_compile_options command instead.
If you still want to use set_target_properties() you may use COMPILE_OPTIONS instead  of COMPILE_FLAGS:
set_target_properties(${TARGET} PROPERTIES COMPILE_OPTIONS ${BUILD_FLAGS})
Since CMake-3.3 (2015)
Anton Petrov suggests to use generator expressions as presented in an answer of ar31. 
The CMake generator expressions applies your ${BUILD_FLAGS} to:
- C++ language using $<COMPILE_LANGUAGE:CXX>(can also beC,CUDA...)
- Clang compiler using $<CXX_COMPILER_ID:Clang>
 (can also beGNUforgcc, orMSVCfor Visual C++... see full list)
 (use$<C_COMPILER_ID:Clang>instead if language is C)
- and more as supported C++ feature or compiler version... (see documentation)
In you case you can use:
target_compile_options(${TARGET} PRIVATE
          $<$<COMPILE_LANGUAGE:CXX>:${BUILD_FLAGS_FOR_CXX}>
          $<$<COMPILE_LANGUAGE:C>:${BUILD_FLAGS_FOR_C}>)
or about compilers:
target_compile_options(${TARGET} PRIVATE
          $<$<CXX_COMPILER_ID:Clang>:${BUILD_FLAGS_FOR_CLANG}>
          $<$<CXX_COMPILER_ID:GNU>:${BUILD_FLAGS_FOR_GCC}>
          $<$<CXX_COMPILER_ID:MSVC>:${BUILD_FLAGS_FOR_VISUAL}>)
Since CMake-3.13 (2018)
A new function target_link_options() allow to pass options to the linker, as mentioned by Craig Scott.
Different options for C and C++ files
The best way is to distinguish C files and C++ files using two different targets.