Each infinite straight wire produces a magnetic field $B = \mu_0 I / (2\pi d)$ at a perpendicular distance $d$.
At the origin (where both wires cross), the distance from each wire to the origin is zero — field would be infinite. In practice this is a limiting geometry.
For wires kept at 90°, the fields due to each wire at any point off origin are perpendicular, and the net field:
$$B_{net} = \sqrt{B_1^2 + B_2^2}$$
If both produce equal magnitudes at a given point: $B_{net} = \sqrt{2}\,B$.
Direction is at 45° to each wire.