This command prints > followed by the contents of the first column, then a new line character (\n) followed by second column. It is a fairly trivial operation and should be easy to find many similar solutions by Googling this site or the whole internet.

awk '{print ">"$1"\n"$2}' input.txt > output.fas

You can try sed 's/\(seq-name[0-9]\)\s\(DNA-sequence[0-9]\)/>\1\n\2/g' input_file > output_file

perl: perl -ae 'print ">$F[0]\n$F[1]\n";' in >out.fa