This answer was made by using the Oxford advanced learner's dictionary, more specifically, the entry for "see" in this dictionary, and its examples (2005 paper edition).
I wish to see my children to have a happy life.
This construction is of the type "V - object - to inf". It is not used for the verb "to see".
"V (passive) - to inf" is used, but only when this verb means "use eyes", "to become aware of something by using your eyes";
- He was seen to enter the building about the time when the crime was committed.
The idea cannot be expressed properly by means of this construction when the verb is "to see". The following sentence makes little sense in a current context.
- I wish to become aware that my children have a happy life.
I wish to see my children having a happy life.
This construction is of the type "V - O - -ing". It is used with "to see", but only according to two senses of "to see":
when "to see" means "use eyes", "to become aware of something by using your eyes" (They saw her running away from the scene of the crime.),
when "to see" means "imagine" (I can't see her changing her mind.).
"V (passive) - to inf" is used, but only when this verb means "use eyes", "to become aware of something by using your eyes" or when it means "to understand";
- The government not only has to do something, it must be seen to be doing something. (It must be understood that it is doing something.)
The first possibility in the OP, if changed by removing "to", is nto better.
- I wish to see my children have a happy life.
This the construction "V - O - inf", and there is just one sense of "to see" for which it can be used; that's when "to see" means "use eyes", "to become aware of something by using your eyes";
- I saw you put the key in your pocket.
There is also the possibility "V - O - adj", and it has been suggested in another answer, but it won't do either. "I wish to see my children happy." would mean "I wish to see my children when they are happy."; for instance,
- "I don't like to see her unhappy.",
means precisely "I don't like to be aware that she is unhappy when that happens.".
It appears that, whichever way you may try to adjust this construction, nothing results in an idiomatic sentence; that is so, of course if you want to preserve the initial idea, which seems, clearly enough, to be that someone wants that their children had a happy life, in the present and most certainly in the future too.
One could use "to see" in a construction that renders a somewhat diverging idea; this would result from the use of the multi-word verb "see to sth".
- I wish to see to my children's happiness in life.
You are then saying that your wishes are to make what is possible so that your children will be happy.
Otherwise, the initial idea is best expressed differently because there is no figurative meaning that associates the idea of seeing literally and that of the enjoyment that results with an abstract concept. Here is a possibility.
- I wish for my children to have a happy life.