Cost sensitive learning and class balancing
Not clear if you are saying the cost of a FN or a FP is higher, you only mention FN in your statement. Think that you mean a FN is more costly and that a positive means a 1.
In general, if an incorrect prediction for the minority case is more costly (FN), you should sample that minority case higher or the majority case lower so the ratio is closer to 1:1. Balancing will help increase the accuracy of your model when predicting the minority case. Accuracy in predicting the majority case will already be higher as there are more samples to use for that case. Undersampling, oversampling and SMOTE are all useful ways to accomplish this balancing of samples, and each has their own strengths and weaknesses.
However, doing this sample balancing will quickly increase the number of FP, so even though the cost may be lower for a FP, the cost will add up quickly. For example, every 1 FP that you decrease, you may get 10 or 20 more FN
After doing this balancing, you can start to adjust the weights to get the best ratio of FN to FP, trying to get the total cost as low as possible.
minimizing: total cost = FN x cost_of_fn + FP x cost_of_fp
Not sure if there is a mathematical equation to solve this, but you can run this iteratively, change weight ratios for the 2 classes, and calculate the total cost using a confusion matrix to get FN and FP, and graph the results for cost (y) vs weight ratio(x), looking for a minima. I would start with a ratio that is equal to the ratio of your costs.
Example: If the cost of a FN is 10 dollars and the cost of a FP is 1 dollar, then the ratio should be 10:1 for minority:majority class
A straightforward calculation shows how to derive the desired mix of class i: μ(i) in terms of the calculated weight w(i) and number of examples in the training set (before any up or down sampling) n(i). This formula is given below:

Reference: https://medium.com/rv-data/how-to-do-cost-sensitive-learning-61848bf4f5e7