The exclusive reactions gamma p ->(K) over bar (0)K(+)n and gamma p ->(K) over bar (0)K(0)p have been studied in the photon energy range 1.6-3.8 GeV, searching for evidence of the exotic baryon Theta(+)(1540) in the decays Theta(+)-> nK(+) and Theta(+)-> pK(0). Data were collected with the CLAS detector at the Thomas Jefferson National Accelerator Facility. The integrated luminosity was about 70 pb(-1). The reactions have been isolated by detecting the K+ and proton directly, the neutral kaon via its decay to K-S ->pi(+)pi(-) and the neutron or neutral kaon via the missing mass technique. The mass and width of known hyperons such as Sigma(+), Sigma(-) and Lambda(1116) were used as a check of the mass determination accuracy and experimental resolution. Approximately 100 000 Lambda(*)(1520)'s and 150 000 phi's were observed in the (K) over bar (0)K(+)n and (K) over bar (0)K(0)p final state, respectively. No evidence for the Theta(+) pentaquark was found in the nK(+) or pK(S) invariant mass spectra. Upper limits were set on the production cross section of the reaction gamma p ->(K) over bar (0)Theta(+) as functions of center-of-mass angle, nK(+) and pK(S) masses. Combining the results of the two reactions, the 95% C.L. upper limit on the total cross section for a resonance peaked at 1540 MeV was found to be 0.7 nb. Within most of the available theoretical models, this corresponds to an upper limit on the Theta(+) width, Gamma(+)(Theta), ranging between 0.01 and 7 MeV.