20-109 - each digit is used 19 times.
200-1099 - each digit is used 280 times
2000-10999 - each digit is used 3700 times
20000-109999 - each digit used 46000 times.
In fact, it looks like the range from 2x10^n to 11x10^n - 1 (n = 0, 1, 2, 3...) will have the desired property.
2 followed by n 0s ... 10 followed by (n-1) 9s uses each digit OEIS 
A081045 number of times
David P Stigant, Kirk Bresniker, Doron Osovlanski, Nathan Stohler, Joseph DeVincentis

The Following Sequence uses all the digits six times: 12233445566778890999, 12233445566778891000, 12233445566778891001.
-Arthur Drehman

The following range, for any n, seems to use each of the digits from 
0 to 9 exactly the same number of times: 2*(10^n) <= k < 11*(10^n)
The example that Bernardo provides is the set for n=0.
Are there any sequences that are substantially different from this 
category?
Kenny Young

10234567890
10234567891
10234567892
10234567893
10234567894
10234567895
10234567896
10234567897
10234567898
10234567899
In general, let N be a single integer that contains each digit (0-9) k
times.  Then (10N+0, 10N+1, ..., 10N+9) is a nicely distributed
sequence of multiplicity 10k+1.  The given sequence is the one with the smallest value of N in this family.
Juha Saukkola, Clinton Weaver, Robert Wainwright, Heinrich Hemme, Jonathan Fero, Corey Maley, Luke Pebody, Siva Naga Prasad Sajja, Bob Kraus, maurizio codogno, Joseph DeVincentis, Steve Amato