The Kantian objection really only has force against the first version not the modal version. It's ironic that the great exegete of the Ontological Argument in the Middle Ages, Scotus, was also an opponent of inflationary takes on Existence.
The Kantian objection claims that statements to the effect 'X exists' do not function like normal subject-predicate sentences in attributing a property I.e. Existence - in other words contra Thomism existence is not something supper-added to an essence. To claim that is well and good but unless it's backed up by further proofs as to why it's the case it amounts to mere question-begging, something Nagasawa along with others points out.
The 'modern' i.e. 20th century version this criticism argues for Existence along Fregean lines as being a second-level predicate of concepts instead of a first level predicate of objects. Nowadays there's a lot more controversy over whether the Fregean Quantifier approach really capture what we mean when we talk of individual existence or if it's even sufficient in its own sphere. There are modern logician such as Colin Mcginn (and Salmon?) who endorse full-scale property takes on Existence.
For a nice account of the above albeit with a slightly different emphasis of Frege's valuable notion of 'marks' this article is worth checking out.
