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TL;DR: At 1000 m depth water is about 0.5 % denser and 1 % less viscous than at the surface, using the NIST chemistry webbook calculator.
First (this result surprised me a bit): How "thick" some fluid is, is something we measure as "viscosity". Fluids with high viscosity are "thick" (like honey) while for example air has very low viscosity. Viscosity is a function of temperature and pressure (or temperature and density, your choice). At surface conditions and 277 K (0.1 MPa and 4^o^ C) water has a viscosity of about 1574.9 micro Pa s, while at 10 MPa and 4^o^ C (1000 m depth) it has a viscosity of about 1559.8 micro Pa s, so about a 1 % decrease. Which means water actually gets less "thick" as you go deeper.
Second: The density of water (how "heavy" it is) varies slightly with pressure and temperature (as mentioned by @sylveon@lemmy.blahaj.zone). For example you might have heard that water is densest at 4^o^ C (at atmospheric pressure). So at the surface, where the pressure is about 0.1 MPa water has a density of about 999.97 kg / m^3^, while at 1000 m depth, where the pressure is about 10 MPa it has a density of about 1004.9 kg / m^3^, about a 0.5 % increase. However, for most practical applications it's perfectly fine to assume that water is incompressible (as mentioned by many others).
Of course, both density and viscosity also depend on salt content, other impurities etc. but including that in computations is orders of magnitude more difficult than doing computations for pure water, and pure water is probably a good indicator. You can check out more for yourself using the NIST chemistry webbook calculator.
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