This notion is very hard to evaluate without looking at a variety of constitutions and at the circumstances in which they were made. In general, I would say that constitutions can be made to protect the elites, but that this is not necessarily the case. The less democratic the procedure for the creation of the constitution, the more likely that it will be made to protect the elites.
In a sense, it is very probable that all constitutions will be made in a way that is acceptable to elites. Any body that comes together to create a constitution is surely going to be made up largely of members of the elite. It is unlikely that the elites would purposely create a constitution that would harm them. This will be even truer if the constitution does not have to be submitted to a true vote of the people. If the people are allowed to vote on the proposed constitution, it will be harder for the elites to construct the document in such a way that it will only be for their own benefit. They will have to make the constitution broadly acceptable in order for it to gain support among the people. However, if the constitution only needs to be voted on by some elite group, the elites can write it in a way that is only/mostly for their benefit.
However, we also need to be aware that the very act of writing a constitution can work as a check on elite power. If the constitution is taken seriously, people from outside the elite can use it for their own purposes. They can use the procedures and ideas set down in the constitution in ways that might not have been what the elites intended. In addition, the act of writing the constitution limits the elites (again, assuming the constitution is actually going to be respected and followed). Without a constitution, they can typically make up ad hoc procedures and rules as they go along. They can change the way things are done when they feel it is beneficial to them. If there is a written constitution that is taken seriously, this becomes more difficult.
The validity of this notion, then, depends on a variety of factors. Chief among these factors are how much voice the people have in creating the constitution and how seriously the constitution will be taken once it has been created. A deeper analysis of the notion would rely on a detailed analysis of the terms of various constitutions and the circumstances in which they were made and adopted.