Java 7, 132 124 bytes

String f(int n,char[]a){String o="";int i;for(char b:a)if(b>32)for(i=0,o+=b;i++<n;o+=(char)(33+Math.random()*94));return o;}

Nothing fancy, just a double loop like you'd expect. Outer to loop the string, inner to fill in randoms:

String f(int n,char[]a){
    String o="";
    int i;
    for(char b:a)
        if(b>32)
            for(i=0,
                o+=b;
                    i++<n;
                        o+=(char)(33+Math.random()*94));
    return o;
}

Python 2, 123 122 118 114 98 Bytes

Man, I wish random wasn't so expensive (and that we didn't have to filter for spaces). Now we have big savings from being allowed to have cipher characters at the end :) Anyways, here ya go:

from random import*
f=lambda s,n:s and(' '<s[0])*eval('s[0]'+'+chr(randint(32,126))'*n)+f(s[1:],n)

05AB1E, 11 bytes

ð-vy²FžQ.RJ

Try it online!

Explanation to come when I'm at a Computer.

Octave, 39 bytes

@(s,n)[s;32+94*rand(n,nnz(s))](1:end-n)

This takes a string s and an integer n as input. A string in Octave is simply an array of characters. It appends a matrix with n rows and the same number of columns as s has, containing floating point numbers between 32 and 126. It is implicitly rounded to integers and converted to ASCII-characters when it's concatenated with the string s. (1:end-n) straightens this to a horizontal array of characters, where the last n elements are removed.

Test it here!

C#, 141 131 bytes

Pretty similar to @Geobit's Java answer, except longer currently :(

(I,n)=>{var R=new System.Random();var o="";int i,r;foreach(var c in I)if(c>32)for(i=r=0,o+=c;i++<n;){r=R.Next(33,127);o+=(char)r;}return o;};

Full lambda stuff:

Func<string, int, string> a = (I,n) =>
{
    var R=new System.Random();
    var o="";
    int i;
    foreach(var c in I)
        if(c>32)
            for(i=0,o+=c;i++<n;o+=(char)R.Next(33,127));
    return o;
};

CJam, 21 18 bytes

lS-(ol~f{{95mrSc+\}*}

Try it online!

Prints n random trailing characters.

Explanation

lS-        e# Read line and remove spaces.
l~         e# Read and evaluate another line.
f{         e# For each character (passing in N)...
  {        e#   Do this N times...
    95mr   e#     Push random integer in [0, 95).
    Sc+    e#     Add to space character, giving a random printable ASCII character.
  }*
}
           e# All characters remaining on the stack are printed implicitly
           e# at the end of the program.

Pyke, 12 11 9 bytes

d-FQV~KHs(<

Try it here!

d-        -  remove spaces from input
  F       - for i in ^:
   QV     -  repeat (number) times:
     ~KH  -    random_from(printable)
        s -   sum(^)

Trailing random characters is fine according to OP.

JavaScript (Firefox 30+), 101 bytes

(s,n)=>s.replace(/. *(?=.)/g,x=>x[0]+String.fromCharCode(...[for(_ of Array(n))Math.random()*95+32]))

Pure ES6 is two bytes longer:

(s,n)=>s.replace(/. *(?=.)/g,x=>x[0]+String.fromCharCode(...[...Array(n)].map(_=>Math.random()*95+32)))

R, 97 bytes

Unnamed function taking inputs x (string) and n.

function(x,n,l=nchar(x))for(i in 1:l)cat(substr(x,i,i),if(i<l)intToUtf8(sample(32:126,n)),sep="")

Try it on R-fiddle

Q/KDB+, 39 36 Bytes

raze{""sv(raze x;`char$40+n?87)}prior s

(,/)({""sv((,/)x;`char$40+n?87)}':)s

Variables in use:

s:"The treasure is here"
n:2

This uses the prior adverb, which applies the function to it's left between each item to the right and its predecessor. (Essentially, applies function to the left between each character on the right.)

Generate n random numbers between 40 and 126 then convert them to a character equivilent: (q seems to only have characters for these)

`char$40+n?87

//Possible characters.
()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\\]^_`abcdefghijklmnopqrstuvwxyz{|}~

Example output:

TVghrveVp Rpti+r0sea3a9nsIjuRXrAReJ; +di=ys`{ ikhKTe4trTZesz

EDIT:
Saved 3 bytes by converting q's raze into (,/) using the k notation and similarly changed prior to `:

Bash, 124 bytes

Pure bash + coreutils, no control flow structures, no sub-languages, no "eval"

Golfed

E() { N=${1// /};paste <(fold -1<<<$N) <(tr -cd "\\40-\\176"<\/dev\/urandom|head -c$(($2*${#N}-$2))|fold -$2)|tr -d '\t\n';}

Test

>E "The treasure is here" 2
TkZhf(e&Rt@FrS,edha+-sJTuh.rX@eVKi+3s<7hftey8r*/e