Kaon photo- and electroproduction off a proton near the production threshold are investigated by utilizing an isobar model. The background amplitude of the model is constructed from Feynman diagrams, whereas the resonance term is calculated by using the multipole formalism. It is found that both pseudoscalar and pseudovector models can nicely describe the available photoproduction data up to W=50 MeV above the threshold. The Λ resonance S01(1800) is found to play an important role in improving the model. In the case of double polarization observables Ox′ and Oz ′ our result corroborates the finding of Sandorfi (arXiv:0912.3505 [nucl-th]). Due to the large contributions of the K * and K1 vector mesons, extending the model to the case of electroproduction is almost impossible unless either special form factors that strongly suppress their contributions are introduced or all hadronic coupling constants are refitted to both photo- and electroproduction databases, simultaneously. It is also concluded that investigation of the kaon electromagnetic form factor is not recommended near the threshold region.