What makes a C standard library function dangerous, and what is the alternative?
Yes, fgets(..., ..., STDIN) is a good alternative to gets(), because it takes a size parameter (gets() has in fact been removed from the C standard entirely in C11). Note that fgets() is not exactly a drop-in replacement for gets(), because the former will include the terminating \n character if there was room in the buffer for a complete line to be read.
scanf() is considered problematic in some cases, rather than straight-out "bad", because if the input doesn't conform to the expected format it can be impossible to recover sensibly (it doesn't let you rewind the input and try again). If you can just give up on badly formatted input, it's useable. A "better" alternative here is to use an input function like fgets() or fgetc() to read chunks of input, then scan it with sscanf() or parse it with string handling functions like strchr() and strtol(). Also see below for a specific problem with the "%s" conversion specifier in scanf().
It's not a standard C function, but the BSD and POSIX function mktemp() is generally impossible to use safely, because there is always a TOCTTOU race condition between testing for the existence of the file and subsequently creating it. mkstemp() or tmpfile() are good replacements.
strncpy() is a slightly tricky function, because it doesn't null-terminate the destination if there was no room for it. Despite the apparently generic name, this function was designed for creating a specific style of string that differs from ordinary C strings - strings stored in a known fixed width field where the null terminator is not required if the string fills the field exactly (original UNIX directory entries were of this style). If you don't have such a situation, you probably should avoid this function.
atoi() can be a bad choice in some situations, because you can't tell when there was an error doing the conversion (e.g., if the number exceeded the range of an int). Use strtol() if this matters to you.
strcpy(), strcat() and sprintf() suffer from a similar problem to gets() - they don't allow you to specify the size of the destination buffer. It's still possible, at least in theory, to use them safely - but you are much better off using strncat() and snprintf() instead (you could use strncpy(), but see above). Do note that whereas the n for snprintf() is the size of the destination buffer, the n for strncat() is the maximum number of characters to append and does not include the null terminator. Another alternative, if you have already calculated the relevant string and buffer sizes, is memmove() or memcpy().
On the same theme, if you use the scanf() family of functions, don't use a plain "%s" - specify the size of the destination e.g. "%200s".
strtok() is generally considered to be evil because it stores state information between calls. Don't try running THAT in a multithreaded environment!
In the old days, most of the string functions had no bounds checking. Of course they couldn't just delete the old functions, or modify their signatures to include an upper bound, that would break compatibility. Now, for almost every one of those functions, there is an alternative "n" version. For example:
strcpy -> strncpy
strlen -> strnlen
strcmp -> strncmp
strcat -> strncat
strdup -> strndup
sprintf -> snprintf
wcscpy -> wcsncpy
wcslen -> wcsnlen
And more.
See also https://github.com/leafsr/gcc-poison which is a project to create a header file that causes gcc to report an error if you use an unsafe function.