References
According to the MathSciNet review (see also MO answers here and here) this paper should do the trick:
Komrakov, B. P. Maximal subalgebras of real Lie algebras and a problem of Sophus Lie. Dokl. Akad. Nauk SSSR 311 (1990), no. 3, 528--532; translation in Soviet Math. Dokl. 41 (1990), no. 2, 269–273 (1991)
Also, it seems that these papers might be helpful, in conjunction with Dykin's paper Maximal subgroups of the classical groups:
Karpelevič, F. I. The simple subalgebras of the real Lie algebras. Trudy Moskov. Mat. Obšč. 4 (1955), 3–112.
Karpelevič, F. I. Classification of the simple subalgebras of the real forms of classical algebras. Doklady Akad. Nauk SSSR (N.S.) 93, (1953). 613–616.
Karpelevič, F. I. Classification of the simple subgroups of the real forms of the group of complex unimodular matrices. Doklady Akad. Nauk SSSR (N.S.) 85, (1952). 1205–1208.
In fact, that is roughly what is claimed to have been done in:
Selim, Taufik Mohamed On maximal subalgebras in classical real Lie algebras. Selected translations. Selecta Math. Soviet. 6 (1987), no. 2, 163–176.
Another paper that might be helpful is:
King, Oliver On some maximal subgroups of the classical groups. J. Algebra 68 (1981), no. 1, 109–120.
Direct Computation?
Here is an idea how to proceed when $G$ is simple. In that case, every maximal closed connected subgroup of positive dimension is parabolic, the normalizer of a connected semisimple subgroup, or the normalizer of a maximal torus. There are finitely many conjugacy classes of parabolics, maximal tori, and by Richardson's A rigidity theorem for subalgebras of Lie and associative algebras there are also finitely many conjugacy classes normalizers of semisimple subgroups. In the cases you are interested in you could try to work out the maximal subgroups in these three classes.
For example, maximal parabolics of $\mathrm{GL}_n$ are addressed here (also see the references therein). And the aforementioned classification of simple subalgebras (by Karpelevič) ought to give a way to classify the maximal subgroups that are normalizers of semisimple subgroups.