Like others have stated, low intensity non-ionizing electromagnetic radiation (the radiation from wires in your home) has no known health effects. That said, it does not hurt to minimize EMI as it allows your electronics and wireless devices to operate more reliably.
Speaking specifically to Ethernet wires; these have one of the lowest radiation of all consumer grade wires because they use a twisted pair of wires that cancels out any emissions. They are designed this way to protect the signal on the wires from external noise, but this effect works both ways. To additionally reduce the emission you can buy shielded Cat5, Cat5e, Cat6, Ethernet cable (Shielded Twisted Pair). The foil around the wire forms a EMI shield further reducing any emissions in or out. And for further reduction you can add ferrite filter around the outside of the cable at the Ethernet switch to reduce "common mode" emission.
To put it into perceptive, the 60hz electrical wires in your home, and FM radio station transmitter on the other side of town have orders of magnitude larger intensities in your home than standard unshielded ethernet wires. Typically the largest EMF emitters in the home are low quality power supplies that are not FCC approved. The FCC sticker on the power supply indicates that the device design has been tested for EMI and complies with United States regulations. You can check the back of your router, modem, and both power supplies for this mark. Most likely they will all have it, but if not I recommend getting a different brand.