Node exports, architecture
That does not look very DRY ( Dont repeat yourself ), not to mention that it does not look good that every function must be aware of the connection string.

For updates, you could do something like this:

exports.setPassword = function (id, password, callback) {
  genericUpdate('UPDATE users SET password=$1 WHERE id=$2', [password, id], callback );
};

exports.setStatus = function (id, status, callback) {
  genericUpdate('UPDATE users SET status=$1 WHERE id=$2', [status, id], callback );
};

function genericUpdate( queryString, parameters, callback )
{
  callback = callback || function(){};
  pg.connect(conString, function (err, client, done) {
    if (err){ 
      callback(err);
    } else {
      client.query(queryString, parameters, function (err) {
        done();
        callback(err || null);
      });
    }
  });
}


I added curly braces and newlines to your if statements, other than that your code seems fine.

EDIT: some thoughts on selects, you probably want a few generic selection functions, if you create one that is meant for selecting on an id, then you can get away with always returning the first data record if any.

exports.getStayInTouch = function (userId, callback) {
  selectById( 'SELECT * FROM stay_in_touch_contacts WHERE user_id=$1' , userId , callback );
};

function selectById( queryString, id, callback )
{
  callback = callback || function(){};
  pg.connect(conString, function (err, client, done) {
    if (err){ 
      callback(err);
    } else {
      client.query(queryString, id, function (err,results) {
        done();
        if(err)
          callback(err)
        else
          callback(null, results.rows[0] );
      });
    }
  });
}