A python solution

Dependencies: Biopython

#!/usr/bin/python2.7

from Bio import SeqIO
import getopt,sys,re


def usage():
    print "Usage: split.py -i <input_scaffold_fasta> -o <output_contig_fasta>"

try:
    options, remainder=getopt.getopt(sys.argv[1:], 'i:o:h')

except getopt.GetoptError as err:
    print str(err)
    usage()
    sys.exit()

for opt, arg in options:
    if opt in ('-i'):
        input_file=arg
    if opt in ('-h'):
        usage()
    sys.exit()
    elif opt in ('-o'):
        output_file=arg

out=open(output_file, 'w')

sequence = ''.join([str(record.seq).strip() for record in SeqIO.parse(input_file, "fasta")])

m=re.sub('[nN]+','\n',sequence).split('\n')

for i in range(0,len(m)):
    out.write('>contig_'+str(i)+'\n')
    out.write(m[i]+'\n')

Usage

[comp]$ python split.py -i input_fasta_file.fa -o output_fasta.fa

Output

>contig_0
ATCGCTGATCGAT
>contig_1
CTAGCTAGCTAGCGTAGCTGCTAG
>contig_2
CGTAGCCGTAGCT

just for fun:

#!/usr/bin/perl

use strict;
use warnings;

$/ = "\n>";

while (<>) {
    s/>//g;
    my ($id, @seq) = split (/\n+/, $_);
    my @slices = split (/N+/, join("", @seq));
    my $num = 0;
    foreach my $seq (@slices) {
        $num++;
        print ">$id\_$num\n";
        while ($seq) {
            print substr($seq, 0, 80), "\n";
            substr($seq, 0, 80) = '';
        }
    }
}

Run:

$ perl split_by_Ns.pl < test.fa
>chr1_1
ATCGCTGATCGAT
>chr1_2
CTAGCTAGCTAGCGTAGCTGCTAG
>chr1_3
CGTAGCCGTAGCT

Here's a way that doesn't use slow Python libraries, works with repeated stretches of Ns, and should be agnostic to Python 2 or 3:

#!/usr/bin/env python

import sys
import re

not_first = False

def process_seq(header, seq):
    # reduce repeated Ns                                                                                                                                                                                                                          
    seq = re.sub(r'[^\w\s]|(.)(?=\1)', '', seq)
    # get indices of Ns                                                                                                                                                                                                                           
    match_idx = 1
    posn_idx = 0
    for match in re.finditer(r'N(.*?)', seq):
        sys.stdout.write("%s_%d\n%s\n" % (header, match_idx, seq[posn_idx:match.start()]))
        posn_idx = match.end()
        match_idx += 1
    sys.stdout.write("%s_%d\n%s\n" % (header, match_idx, seq[posn_idx:]))

for line in sys.stdin:
    if line.startswith('>'):
        if not_first:
            process_seq(header, seq)
        header = line.rstrip()
        seq = ''
        not_first = True
    else:
        seq += line.rstrip()

if not_first:
    process_seq(header, seq)

Output:

$ ./split_on_Ns.py < input.fa
>chr1_1
ATCGCTGATCGAT
>chr1_2
CTAGCTAGCTAGCGTAGCTGCTAG
>chr1_3
CGTAGCGTAGC

using awk: can handle multiple Ns, case insensitive for Ns:

$  awk '{gsub("[Nn]+","\n>\n");}1' test.fa | awk '/^>/ {if ($0 == ">") {$0=prev} prev=$0}1' | awk '/^>/ {getline seq} {if(seq!="") {print $0"\n"seq}}' | awk '(/^>/ && a[$0]++) {$0=$0"_"a[$0]}1' 
>chr1
ATCGCTGATCGAT
>chr1_2
CTAGCTAGCTAGCGTAGCTGCTAG
>chr1_3
CGTAGCCGTAGCT
>chr2
CCTGA
>chr2_2
GTAGT
>chr3
ATG
>chr4
C
>chr4_2
T

input: $ cat test.fa

>chr1
ATCGCTGATCGATNCTAGCTAGCTAGCGTAGCTGCTAGNCGTAGCCGTAGCT
>chr2
CCTGANGTAGT
>chr3
ATGNNNNNN
>chr4
NNNNCnT

make sure that sequences are single line. If they are not, use following command instead of above line:

$ seqkit seq -w0 test.fa | awk '{gsub("[Nn]+","\n>\n");}1' | awk '/^>/ {if ($0 == ">") {$0=prev} prev=$0}1' | awk '/^>/ {getline seq} {if(seq!="") {print $0"\n"seq}}' | awk '(/^>/ && a[$0]++) {$0=$0"_"a[$0]}1'

Download seqkit from here.