You can align the 1st element of the kernel with each element, and pad with 0s to get a list of the same length as the original:
In[1]:= test = {0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0};
In[2]:= ideal = {1, 1, 1};
In[3]:= wantedResult = {0, 0, 0, 0, 0, 0, 1, 2, 3, 2, 1, 0, 0, 0, 0,
0, 0};
In[4]:= result = ListCorrelate[ideal, test, 1, 0]
Out[4]= {0, 0, 0, 0, 0, 0, 1, 2, 3, 2, 1, 0, 0, 0, 0, 0, 0}
In[5]:= wantedResult === result
Out[5]= True