- Portable across gcc/clang/ICC/MSVC, C and C++.
- fully safe with all optimization levels: no strict-aliasing violation UB
- print in hex as u8, u16, u32, or u64 elements (based on @AG1's answer)
- Prints in memory order (least-significant element first, like _mm_setr_epiX).  Reverse the array indices if you prefer printing in the same order Intel's manuals use, where the most significant element is on the left (like_mm_set_epiX).  Related: Convention for displaying vector registers
Using a __m128i* to load from an array of int is safe because the __m128 types are defined to allow aliasing just like ISO C unsigned char*.  (e.g. in gcc's headers, the definition includes __attribute__((may_alias)).)
The reverse isn't safe (pointing an int* onto part of a __m128i object).  MSVC guarantees that's safe, but GCC/clang don't.  (-fstrict-aliasing is on by default).  It sometimes works with GCC/clang, but why risk it?  It sometimes even interferes with optimization; see this Q&A.  See also Is `reinterpret_cast`ing between hardware SIMD vector pointer and the corresponding type an undefined behavior?
See GCC AVX _m256i cast to int array leads to wrong values for a real-world example of GCC breaking code which points an int* at a __m256i.
(uint32_t*) &my_vector violates the C and C++ aliasing rules, and is not guaranteed to work the way you'd expect.  Storing to a local array and then accessing it is guaranteed to be safe.  It even optimizes away with most compilers, so you get movq / pextrq directly from xmm to integer registers instead of an actual store/reload, for example.
Source + asm output on the Godbolt compiler explorer: proof it compiles with MSVC and so on.
#include <immintrin.h>
#include <stdint.h>
#include <stdio.h>
#ifndef __cplusplus
#include <stdalign.h>   // C11 defines _Alignas().  This header defines alignas()
#endif
void p128_hex_u8(__m128i in) {
    alignas(16) uint8_t v[16];
    _mm_store_si128((__m128i*)v, in);
    printf("v16_u8: %x %x %x %x | %x %x %x %x | %x %x %x %x | %x %x %x %x\n",
           v[0], v[1],  v[2],  v[3],  v[4],  v[5],  v[6],  v[7],
           v[8], v[9], v[10], v[11], v[12], v[13], v[14], v[15]);
}
void p128_hex_u16(__m128i in) {
    alignas(16) uint16_t v[8];
    _mm_store_si128((__m128i*)v, in);
    printf("v8_u16: %x %x %x %x,  %x %x %x %x\n", v[0], v[1], v[2], v[3], v[4], v[5], v[6], v[7]);
}
void p128_hex_u32(__m128i in) {
    alignas(16) uint32_t v[4];
    _mm_store_si128((__m128i*)v, in);
    printf("v4_u32: %x %x %x %x\n", v[0], v[1], v[2], v[3]);
}
void p128_hex_u64(__m128i in) {
    alignas(16) unsigned long long v[2];  // uint64_t might give format-string warnings with %llx; it's just long in some ABIs
    _mm_store_si128((__m128i*)v, in);
    printf("v2_u64: %llx %llx\n", v[0], v[1]);
}
If you need portability to C99 or C++03 or earlier (i.e. without C11 / C++11), remove the alignas() and use storeu instead of store.  Or use __attribute__((aligned(16))) or __declspec( align(16) ) instead.
(If you're writing code with intrinsics, you should be using a recent compiler version.  Newer compilers usually make better asm than older compilers, including for SSE/AVX intrinsics.  But maybe you want to use gcc-6.3 with -std=gnu++03 C++03 mode for a codebase that isn't ready for C++11 or something.)
Sample output from calling all 4 functions on
// source used:
__m128i vec = _mm_setr_epi8(1, 2, 3, 4, 5, 6, 7,
                            8, 9, 10, 11, 12, 13, 14, 15, 16);
// output:
v2_u64: 0x807060504030201 0x100f0e0d0c0b0a09
v4_u32: 0x4030201 0x8070605 0xc0b0a09 0x100f0e0d
v8_u16: 0x201 0x403 0x605 0x807  | 0xa09 0xc0b 0xe0d 0x100f
v16_u8: 0x1 0x2 0x3 0x4 | 0x5 0x6 0x7 0x8 | 0x9 0xa 0xb 0xc | 0xd 0xe 0xf 0x10
Adjust the format strings if you want to pad with leading zeros for consistent output width.  See printf(3).