My data is is mostly continuous but has one binary variable. I tried the pam algorithm in R with the Gower index, but the number of clusters that give the best silhouette width is 2 – allowing the binary variable to completely dominate the result.
If the binary variable is not very useful, try putting less weight on it.
There is nothing wrong with having a domain expert manually assign weights to different attributes to help the algorithm find new information. That the binary attribute splits the data into two is a correct result, now you want to find something new, so either remove it (weight 0) or at least reduce the weight.
To better justify the chosen number of clusters (k) you can use other partition quality indices than Silhouette width. Example indices based on arbitrary dissimilarity are: Caliński & Harabasz index (chosen as the best in Milligan and Cooper study 1985 and as 4th best in Dimitriadou et al. 2002) generalized for dissimilarities, Dunn index, Gamma index, C index etc. A selection of these and other quality indices is provided by e.g. R's cluster.stats function included in fpc package.
It is common to then choose the k returned by majority of the computed indices, as the final one.
